Article ENUMSET1, MML version 4.99.1005
:: ENUMSET1:funcnot 1 => ENUMSET1:func 1
definition
let a1, a2, a3 be set;
func {A1,A2,A3} -> set means
for b1 being set holds
b1 in it
iff
(b1 <> a1 & b1 <> a2 implies b1 = a3);
end;
:: ENUMSET1:def 1
theorem
for b1, b2, b3, b4 being set holds
b4 = {b1,b2,b3}
iff
for b5 being set holds
b5 in b4
iff
(b5 <> b1 & b5 <> b2 implies b5 = b3);
:: ENUMSET1:funcnot 2 => ENUMSET1:func 2
definition
let a1, a2, a3, a4 be set;
func {A1,A2,A3,A4} -> set means
for b1 being set holds
b1 in it
iff
(b1 <> a1 & b1 <> a2 & b1 <> a3 implies b1 = a4);
end;
:: ENUMSET1:def 2
theorem
for b1, b2, b3, b4, b5 being set holds
b5 = {b1,b2,b3,b4}
iff
for b6 being set holds
b6 in b5
iff
(b6 <> b1 & b6 <> b2 & b6 <> b3 implies b6 = b4);
:: ENUMSET1:funcnot 3 => ENUMSET1:func 3
definition
let a1, a2, a3, a4, a5 be set;
func {A1,A2,A3,A4,A5} -> set means
for b1 being set holds
b1 in it
iff
(b1 <> a1 & b1 <> a2 & b1 <> a3 & b1 <> a4 implies b1 = a5);
end;
:: ENUMSET1:def 3
theorem
for b1, b2, b3, b4, b5, b6 being set holds
b6 = {b1,b2,b3,b4,b5}
iff
for b7 being set holds
b7 in b6
iff
(b7 <> b1 & b7 <> b2 & b7 <> b3 & b7 <> b4 implies b7 = b5);
:: ENUMSET1:funcnot 4 => ENUMSET1:func 4
definition
let a1, a2, a3, a4, a5, a6 be set;
func {A1,A2,A3,A4,A5,A6} -> set means
for b1 being set holds
b1 in it
iff
(b1 <> a1 & b1 <> a2 & b1 <> a3 & b1 <> a4 & b1 <> a5 implies b1 = a6);
end;
:: ENUMSET1:def 4
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
b7 = {b1,b2,b3,b4,b5,b6}
iff
for b8 being set holds
b8 in b7
iff
(b8 <> b1 & b8 <> b2 & b8 <> b3 & b8 <> b4 & b8 <> b5 implies b8 = b6);
:: ENUMSET1:funcnot 5 => ENUMSET1:func 5
definition
let a1, a2, a3, a4, a5, a6, a7 be set;
func {A1,A2,A3,A4,A5,A6,A7} -> set means
for b1 being set holds
b1 in it
iff
(b1 <> a1 & b1 <> a2 & b1 <> a3 & b1 <> a4 & b1 <> a5 & b1 <> a6 implies b1 = a7);
end;
:: ENUMSET1:def 5
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
b8 = {b1,b2,b3,b4,b5,b6,b7}
iff
for b9 being set holds
b9 in b8
iff
(b9 <> b1 & b9 <> b2 & b9 <> b3 & b9 <> b4 & b9 <> b5 & b9 <> b6 implies b9 = b7);
:: ENUMSET1:funcnot 6 => ENUMSET1:func 6
definition
let a1, a2, a3, a4, a5, a6, a7, a8 be set;
func {A1,A2,A3,A4,A5,A6,A7,A8} -> set means
for b1 being set holds
b1 in it
iff
(b1 <> a1 & b1 <> a2 & b1 <> a3 & b1 <> a4 & b1 <> a5 & b1 <> a6 & b1 <> a7 implies b1 = a8);
end;
:: ENUMSET1:def 6
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
b9 = {b1,b2,b3,b4,b5,b6,b7,b8}
iff
for b10 being set holds
b10 in b9
iff
(b10 <> b1 & b10 <> b2 & b10 <> b3 & b10 <> b4 & b10 <> b5 & b10 <> b6 & b10 <> b7 implies b10 = b8);
:: ENUMSET1:funcnot 7 => ENUMSET1:func 7
definition
let a1, a2, a3, a4, a5, a6, a7, a8, a9 be set;
func {A1,A2,A3,A4,A5,A6,A7,A8,A9} -> set means
for b1 being set holds
b1 in it
iff
(b1 <> a1 & b1 <> a2 & b1 <> a3 & b1 <> a4 & b1 <> a5 & b1 <> a6 & b1 <> a7 & b1 <> a8 implies b1 = a9);
end;
:: ENUMSET1:def 7
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being set holds
b10 = {b1,b2,b3,b4,b5,b6,b7,b8,b9}
iff
for b11 being set holds
b11 in b10
iff
(b11 <> b1 & b11 <> b2 & b11 <> b3 & b11 <> b4 & b11 <> b5 & b11 <> b6 & b11 <> b7 & b11 <> b8 implies b11 = b9);
:: ENUMSET1:funcnot 8 => ENUMSET1:func 8
definition
let a1, a2, a3, a4, a5, a6, a7, a8, a9, a10 be set;
func {A1,A2,A3,A4,A5,A6,A7,A8,A9,A10} -> set means
for b1 being set holds
b1 in it
iff
(b1 <> a1 & b1 <> a2 & b1 <> a3 & b1 <> a4 & b1 <> a5 & b1 <> a6 & b1 <> a7 & b1 <> a8 & b1 <> a9 implies b1 = a10);
end;
:: ENUMSET1:def 8
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
b11 = {b1,b2,b3,b4,b5,b6,b7,b8,b9,b10}
iff
for b12 being set holds
b12 in b11
iff
(b12 <> b1 & b12 <> b2 & b12 <> b3 & b12 <> b4 & b12 <> b5 & b12 <> b6 & b12 <> b7 & b12 <> b8 & b12 <> b9 implies b12 = b10);
:: ENUMSET1:th 41
theorem
for b1, b2 being set holds
{b1,b2} = {b1} \/ {b2};
:: ENUMSET1:th 42
theorem
for b1, b2, b3 being set holds
{b1,b2,b3} = {b1} \/ {b2,b3};
:: ENUMSET1:th 43
theorem
for b1, b2, b3 being set holds
{b1,b2,b3} = {b1,b2} \/ {b3};
:: ENUMSET1:th 44
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b1} \/ {b2,b3,b4};
:: ENUMSET1:th 45
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b1,b2} \/ {b3,b4};
:: ENUMSET1:th 46
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b1,b2,b3} \/ {b4};
:: ENUMSET1:th 47
theorem
for b1, b2, b3, b4, b5 being set holds
{b1,b2,b3,b4,b5} = {b1} \/ {b2,b3,b4,b5};
:: ENUMSET1:th 48
theorem
for b1, b2, b3, b4, b5 being set holds
{b1,b2,b3,b4,b5} = {b1,b2} \/ {b3,b4,b5};
:: ENUMSET1:th 49
theorem
for b1, b2, b3, b4, b5 being set holds
{b1,b2,b3,b4,b5} = {b1,b2,b3} \/ {b4,b5};
:: ENUMSET1:th 50
theorem
for b1, b2, b3, b4, b5 being set holds
{b1,b2,b3,b4,b5} = {b1,b2,b3,b4} \/ {b5};
:: ENUMSET1:th 51
theorem
for b1, b2, b3, b4, b5, b6 being set holds
{b1,b2,b3,b4,b5,b6} = {b1} \/ {b2,b3,b4,b5,b6};
:: ENUMSET1:th 52
theorem
for b1, b2, b3, b4, b5, b6 being set holds
{b1,b2,b3,b4,b5,b6} = {b1,b2} \/ {b3,b4,b5,b6};
:: ENUMSET1:th 53
theorem
for b1, b2, b3, b4, b5, b6 being set holds
{b1,b2,b3,b4,b5,b6} = {b1,b2,b3} \/ {b4,b5,b6};
:: ENUMSET1:th 54
theorem
for b1, b2, b3, b4, b5, b6 being set holds
{b1,b2,b3,b4,b5,b6} = {b1,b2,b3,b4} \/ {b5,b6};
:: ENUMSET1:th 55
theorem
for b1, b2, b3, b4, b5, b6 being set holds
{b1,b2,b3,b4,b5,b6} = {b1,b2,b3,b4,b5} \/ {b6};
:: ENUMSET1:th 56
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
{b1,b2,b3,b4,b5,b6,b7} = {b1} \/ {b2,b3,b4,b5,b6,b7};
:: ENUMSET1:th 57
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
{b1,b2,b3,b4,b5,b6,b7} = {b1,b2} \/ {b3,b4,b5,b6,b7};
:: ENUMSET1:th 58
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
{b1,b2,b3,b4,b5,b6,b7} = {b1,b2,b3} \/ {b4,b5,b6,b7};
:: ENUMSET1:th 59
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
{b1,b2,b3,b4,b5,b6,b7} = {b1,b2,b3,b4} \/ {b5,b6,b7};
:: ENUMSET1:th 60
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
{b1,b2,b3,b4,b5,b6,b7} = {b1,b2,b3,b4,b5} \/ {b6,b7};
:: ENUMSET1:th 61
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
{b1,b2,b3,b4,b5,b6,b7} = {b1,b2,b3,b4,b5,b6} \/ {b7};
:: ENUMSET1:th 62
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8} = {b1} \/ {b2,b3,b4,b5,b6,b7,b8};
:: ENUMSET1:th 63
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8} = {b1,b2} \/ {b3,b4,b5,b6,b7,b8};
:: ENUMSET1:th 64
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8} = {b1,b2,b3} \/ {b4,b5,b6,b7,b8};
:: ENUMSET1:th 65
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8} = {b1,b2,b3,b4} \/ {b5,b6,b7,b8};
:: ENUMSET1:th 66
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8} = {b1,b2,b3,b4,b5} \/ {b6,b7,b8};
:: ENUMSET1:th 67
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8} = {b1,b2,b3,b4,b5,b6} \/ {b7,b8};
:: ENUMSET1:th 68
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8} = {b1,b2,b3,b4,b5,b6,b7} \/ {b8};
:: ENUMSET1:th 69
theorem
for b1 being set holds
{b1,b1} = {b1};
:: ENUMSET1:th 70
theorem
for b1, b2 being set holds
{b1,b1,b2} = {b1,b2};
:: ENUMSET1:th 71
theorem
for b1, b2, b3 being set holds
{b1,b1,b2,b3} = {b1,b2,b3};
:: ENUMSET1:th 72
theorem
for b1, b2, b3, b4 being set holds
{b1,b1,b2,b3,b4} = {b1,b2,b3,b4};
:: ENUMSET1:th 73
theorem
for b1, b2, b3, b4, b5 being set holds
{b1,b1,b2,b3,b4,b5} = {b1,b2,b3,b4,b5};
:: ENUMSET1:th 74
theorem
for b1, b2, b3, b4, b5, b6 being set holds
{b1,b1,b2,b3,b4,b5,b6} = {b1,b2,b3,b4,b5,b6};
:: ENUMSET1:th 75
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
{b1,b1,b2,b3,b4,b5,b6,b7} = {b1,b2,b3,b4,b5,b6,b7};
:: ENUMSET1:th 76
theorem
for b1 being set holds
{b1,b1,b1} = {b1};
:: ENUMSET1:th 77
theorem
for b1, b2 being set holds
{b1,b1,b1,b2} = {b1,b2};
:: ENUMSET1:th 78
theorem
for b1, b2, b3 being set holds
{b1,b1,b1,b2,b3} = {b1,b2,b3};
:: ENUMSET1:th 79
theorem
for b1, b2, b3, b4 being set holds
{b1,b1,b1,b2,b3,b4} = {b1,b2,b3,b4};
:: ENUMSET1:th 80
theorem
for b1, b2, b3, b4, b5 being set holds
{b1,b1,b1,b2,b3,b4,b5} = {b1,b2,b3,b4,b5};
:: ENUMSET1:th 81
theorem
for b1, b2, b3, b4, b5, b6 being set holds
{b1,b1,b1,b2,b3,b4,b5,b6} = {b1,b2,b3,b4,b5,b6};
:: ENUMSET1:th 82
theorem
for b1 being set holds
{b1,b1,b1,b1} = {b1};
:: ENUMSET1:th 83
theorem
for b1, b2 being set holds
{b1,b1,b1,b1,b2} = {b1,b2};
:: ENUMSET1:th 84
theorem
for b1, b2, b3 being set holds
{b1,b1,b1,b1,b2,b3} = {b1,b2,b3};
:: ENUMSET1:th 85
theorem
for b1, b2, b3, b4 being set holds
{b1,b1,b1,b1,b2,b3,b4} = {b1,b2,b3,b4};
:: ENUMSET1:th 86
theorem
for b1, b2, b3, b4, b5 being set holds
{b1,b1,b1,b1,b2,b3,b4,b5} = {b1,b2,b3,b4,b5};
:: ENUMSET1:th 87
theorem
for b1 being set holds
{b1,b1,b1,b1,b1} = {b1};
:: ENUMSET1:th 88
theorem
for b1, b2 being set holds
{b1,b1,b1,b1,b1,b2} = {b1,b2};
:: ENUMSET1:th 89
theorem
for b1, b2, b3 being set holds
{b1,b1,b1,b1,b1,b2,b3} = {b1,b2,b3};
:: ENUMSET1:th 90
theorem
for b1, b2, b3, b4 being set holds
{b1,b1,b1,b1,b1,b2,b3,b4} = {b1,b2,b3,b4};
:: ENUMSET1:th 91
theorem
for b1 being set holds
{b1,b1,b1,b1,b1,b1} = {b1};
:: ENUMSET1:th 92
theorem
for b1, b2 being set holds
{b1,b1,b1,b1,b1,b1,b2} = {b1,b2};
:: ENUMSET1:th 93
theorem
for b1, b2, b3 being set holds
{b1,b1,b1,b1,b1,b1,b2,b3} = {b1,b2,b3};
:: ENUMSET1:th 94
theorem
for b1 being set holds
{b1,b1,b1,b1,b1,b1,b1} = {b1};
:: ENUMSET1:th 95
theorem
for b1, b2 being set holds
{b1,b1,b1,b1,b1,b1,b1,b2} = {b1,b2};
:: ENUMSET1:th 96
theorem
for b1 being set holds
{b1,b1,b1,b1,b1,b1,b1,b1} = {b1};
:: ENUMSET1:th 98
theorem
for b1, b2, b3 being set holds
{b1,b2,b3} = {b1,b3,b2};
:: ENUMSET1:th 99
theorem
for b1, b2, b3 being set holds
{b1,b2,b3} = {b2,b1,b3};
:: ENUMSET1:th 100
theorem
for b1, b2, b3 being set holds
{b1,b2,b3} = {b2,b3,b1};
:: ENUMSET1:th 102
theorem
for b1, b2, b3 being set holds
{b1,b2,b3} = {b3,b2,b1};
:: ENUMSET1:th 103
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b1,b2,b4,b3};
:: ENUMSET1:th 104
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b1,b3,b2,b4};
:: ENUMSET1:th 105
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b1,b3,b4,b2};
:: ENUMSET1:th 107
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b1,b4,b3,b2};
:: ENUMSET1:th 108
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b2,b1,b3,b4};
:: ENUMSET1:th 109
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b2,b1,b4,b3};
:: ENUMSET1:th 110
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b2,b3,b1,b4};
:: ENUMSET1:th 111
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b2,b3,b4,b1};
:: ENUMSET1:th 112
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b2,b4,b1,b3};
:: ENUMSET1:th 113
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b2,b4,b3,b1};
:: ENUMSET1:th 116
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b3,b2,b1,b4};
:: ENUMSET1:th 117
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b3,b2,b4,b1};
:: ENUMSET1:th 118
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b3,b4,b1,b2};
:: ENUMSET1:th 119
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b3,b4,b2,b1};
:: ENUMSET1:th 123
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b4,b2,b3,b1};
:: ENUMSET1:th 125
theorem
for b1, b2, b3, b4 being set holds
{b1,b2,b3,b4} = {b4,b3,b2,b1};
:: ENUMSET1:th 127
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1} \/ {b2,b3,b4,b5,b6,b7,b8,b9};
:: ENUMSET1:th 128
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2} \/ {b3,b4,b5,b6,b7,b8,b9};
:: ENUMSET1:th 129
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2,b3} \/ {b4,b5,b6,b7,b8,b9};
:: ENUMSET1:th 130
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2,b3,b4} \/ {b5,b6,b7,b8,b9};
:: ENUMSET1:th 131
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2,b3,b4,b5} \/ {b6,b7,b8,b9};
:: ENUMSET1:th 132
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2,b3,b4,b5,b6} \/ {b7,b8,b9};
:: ENUMSET1:th 133
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2,b3,b4,b5,b6,b7} \/ {b8,b9};
:: ENUMSET1:th 134
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2,b3,b4,b5,b6,b7,b8} \/ {b9};
:: ENUMSET1:th 135
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9,b10} = {b1,b2,b3,b4,b5,b6,b7,b8,b9} \/ {b10};
:: ENUMSET1:th 136
theorem
for b1, b2, b3 being set
st b1 <> b2 & b1 <> b3
holds {b1,b2,b3} \ {b1} = {b2,b3};
:: ENUMSET1:th 137
theorem
for b1, b2, b3 being set holds
{b2,b1} \/ {b3,b1} = {b1,b2,b3};