Step | Hyp | Ref
| Expression |
1 | | df-addcfn 5746 |
. . 3
⊢ AddC = (x ∈ V, y ∈ V ↦ (x +c y)) |
2 | | elin 3219 |
. . . . . . . 8
⊢ (〈{b}, 〈{z}, 〈x, y〉〉〉 ∈ ( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c)) ↔ (〈{b}, 〈{z}, 〈x, y〉〉〉 ∈ Ins2 Ins2 S ∧ 〈{b}, 〈{z}, 〈x, y〉〉〉 ∈ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c))) |
3 | | snex 4111 |
. . . . . . . . . . 11
⊢ {z} ∈
V |
4 | 3 | otelins2 5791 |
. . . . . . . . . 10
⊢ (〈{b}, 〈{z}, 〈x, y〉〉〉 ∈ Ins2 Ins2 S ↔ 〈{b}, 〈x, y〉〉 ∈ Ins2 S ) |
5 | | vex 2862 |
. . . . . . . . . . 11
⊢ x ∈
V |
6 | 5 | otelins2 5791 |
. . . . . . . . . 10
⊢ (〈{b}, 〈x, y〉〉 ∈ Ins2 S ↔ 〈{b}, y〉 ∈ S
) |
7 | | vex 2862 |
. . . . . . . . . . 11
⊢ b ∈
V |
8 | | vex 2862 |
. . . . . . . . . . 11
⊢ y ∈
V |
9 | 7, 8 | opelssetsn 4760 |
. . . . . . . . . 10
⊢ (〈{b}, y〉 ∈ S ↔ b ∈ y) |
10 | 4, 6, 9 | 3bitri 262 |
. . . . . . . . 9
⊢ (〈{b}, 〈{z}, 〈x, y〉〉〉 ∈ Ins2 Ins2 S ↔ b ∈ y) |
11 | 8 | oqelins4 5794 |
. . . . . . . . . 10
⊢ (〈{b}, 〈{z}, 〈x, y〉〉〉 ∈ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c) ↔ 〈{b}, 〈{z}, x〉〉 ∈ (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c)) |
12 | | elin 3219 |
. . . . . . . . . . . . 13
⊢ (〈{a}, 〈{b}, 〈{z}, x〉〉〉 ∈ ( Ins2 Ins2 S ∩ Ins4 SI3
( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) ↔
(〈{a},
〈{b},
〈{z},
x〉〉〉 ∈ Ins2 Ins2 S ∧ 〈{a}, 〈{b}, 〈{z}, x〉〉〉 ∈ Ins4 SI3
( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup
)))) |
13 | | snex 4111 |
. . . . . . . . . . . . . . . 16
⊢ {b} ∈
V |
14 | 13 | otelins2 5791 |
. . . . . . . . . . . . . . 15
⊢ (〈{a}, 〈{b}, 〈{z}, x〉〉〉 ∈ Ins2 Ins2 S ↔ 〈{a}, 〈{z}, x〉〉 ∈ Ins2 S ) |
15 | 3 | otelins2 5791 |
. . . . . . . . . . . . . . 15
⊢ (〈{a}, 〈{z}, x〉〉 ∈ Ins2 S ↔ 〈{a}, x〉 ∈ S
) |
16 | | vex 2862 |
. . . . . . . . . . . . . . . 16
⊢ a ∈
V |
17 | 16, 5 | opelssetsn 4760 |
. . . . . . . . . . . . . . 15
⊢ (〈{a}, x〉 ∈ S ↔ a ∈ x) |
18 | 14, 15, 17 | 3bitri 262 |
. . . . . . . . . . . . . 14
⊢ (〈{a}, 〈{b}, 〈{z}, x〉〉〉 ∈ Ins2 Ins2 S ↔ a ∈ x) |
19 | 5 | oqelins4 5794 |
. . . . . . . . . . . . . . 15
⊢ (〈{a}, 〈{b}, 〈{z}, x〉〉〉 ∈ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup )) ↔
〈{a},
〈{b},
{z}〉〉 ∈ SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup
))) |
20 | | vex 2862 |
. . . . . . . . . . . . . . . 16
⊢ z ∈
V |
21 | 16, 7, 20 | otsnelsi3 5805 |
. . . . . . . . . . . . . . 15
⊢ (〈{a}, 〈{b}, {z}〉〉 ∈ SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup )) ↔
〈a, 〈b, z〉〉 ∈ ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup
))) |
22 | | elin 3219 |
. . . . . . . . . . . . . . . 16
⊢ (〈a, 〈b, z〉〉 ∈ ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup )) ↔
(〈a,
〈b,
z〉〉 ∈ Ins3 Disj ∧ 〈a, 〈b, z〉〉 ∈ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))
“ Cup ))) |
23 | 20 | otelins3 5792 |
. . . . . . . . . . . . . . . . . 18
⊢ (〈a, 〈b, z〉〉 ∈ Ins3 Disj ↔ 〈a, b〉 ∈ Disj
) |
24 | | df-br 4640 |
. . . . . . . . . . . . . . . . . 18
⊢ (a Disj b ↔ 〈a, b〉 ∈ Disj ) |
25 | 16, 7 | brdisj 5822 |
. . . . . . . . . . . . . . . . . 18
⊢ (a Disj b ↔ (a
∩ b) = ∅) |
26 | 23, 24, 25 | 3bitr2i 264 |
. . . . . . . . . . . . . . . . 17
⊢ (〈a, 〈b, z〉〉 ∈ Ins3 Disj ↔ (a ∩ b) =
∅) |
27 | | trtxp 5781 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (p((2nd ∘ 1st ) ⊗ 2nd )〈b, z〉 ↔
(p(2nd ∘ 1st )b ∧ p2nd z)) |
28 | 27 | anbi2i 675 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((p(1st ∘ 1st )a ∧ p((2nd ∘ 1st ) ⊗ 2nd )〈b, z〉) ↔
(p(1st ∘ 1st )a ∧ (p(2nd ∘ 1st )b ∧ p2nd z))) |
29 | | trtxp 5781 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))〈a, 〈b, z〉〉 ↔ (p(1st ∘ 1st )a ∧ p((2nd ∘ 1st ) ⊗ 2nd )〈b, z〉)) |
30 | | anass 630 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((p(1st ∘ 1st )a ∧ p(2nd ∘ 1st )b) ∧ p2nd z) ↔ (p(1st ∘ 1st )a ∧ (p(2nd ∘ 1st )b ∧ p2nd z))) |
31 | 28, 29, 30 | 3bitr4i 268 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))〈a, 〈b, z〉〉 ↔ ((p(1st ∘ 1st )a ∧ p(2nd ∘ 1st )b) ∧ p2nd z)) |
32 | | brco 4883 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (p(1st ∘ 1st )a ↔ ∃x(p1st x ∧ x1st a)) |
33 | 16 | br1st 4858 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (x1st a ↔ ∃y x = 〈a, y〉) |
34 | 33 | anbi2i 675 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((p1st x ∧ x1st a) ↔ (p1st x ∧ ∃y x = 〈a, y〉)) |
35 | | 19.42v 1905 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (∃y(p1st x ∧ x = 〈a, y〉) ↔ (p1st x ∧ ∃y x = 〈a, y〉)) |
36 | 34, 35 | bitr4i 243 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((p1st x ∧ x1st a) ↔ ∃y(p1st x ∧ x = 〈a, y〉)) |
37 | 36 | exbii 1582 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (∃x(p1st x ∧ x1st a) ↔ ∃x∃y(p1st x ∧ x = 〈a, y〉)) |
38 | | excom 1741 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (∃x∃y(p1st x ∧ x = 〈a, y〉) ↔ ∃y∃x(p1st x ∧ x = 〈a, y〉)) |
39 | | exancom 1586 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (∃x(p1st x ∧ x = 〈a, y〉) ↔ ∃x(x = 〈a, y〉 ∧ p1st x)) |
40 | 16, 8 | opex 4588 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ 〈a, y〉 ∈ V |
41 | | breq2 4643 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (x = 〈a, y〉 → (p1st x ↔ p1st 〈a, y〉)) |
42 | 40, 41 | ceqsexv 2894 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (∃x(x = 〈a, y〉 ∧ p1st x) ↔ p1st 〈a, y〉) |
43 | 39, 42 | bitri 240 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (∃x(p1st x ∧ x = 〈a, y〉) ↔ p1st 〈a, y〉) |
44 | 43 | exbii 1582 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (∃y∃x(p1st x ∧ x = 〈a, y〉) ↔ ∃y p1st 〈a, y〉) |
45 | 38, 44 | bitri 240 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (∃x∃y(p1st x ∧ x = 〈a, y〉) ↔ ∃y p1st 〈a, y〉) |
46 | 32, 37, 45 | 3bitri 262 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (p(1st ∘ 1st )a ↔ ∃y p1st 〈a, y〉) |
47 | 46 | anbi1i 676 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((p(1st ∘ 1st )a ∧ p(2nd ∘ 1st )b) ↔ (∃y p1st 〈a, y〉 ∧ p(2nd ∘ 1st )b)) |
48 | | 19.41v 1901 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (∃y(p1st 〈a, y〉 ∧ p(2nd ∘ 1st )b) ↔ (∃y p1st 〈a, y〉 ∧ p(2nd ∘ 1st )b)) |
49 | 40 | br1st 4858 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (p1st 〈a, y〉 ↔ ∃z p = 〈〈a, y〉, z〉) |
50 | | breq1 4642 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (p = 〈〈a, y〉, z〉 →
(p(2nd ∘ 1st )b ↔ 〈〈a, y〉, z〉(2nd
∘ 1st )b)) |
51 | 40, 20 | brco1st 5777 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (〈〈a, y〉, z〉(2nd ∘ 1st )b ↔ 〈a, y〉2nd b) |
52 | 16, 8 | opbr2nd 5502 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (〈a, y〉2nd
b ↔ y = b) |
53 | 51, 52 | bitri 240 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (〈〈a, y〉, z〉(2nd ∘ 1st )b ↔ y =
b) |
54 | 50, 53 | syl6bb 252 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (p = 〈〈a, y〉, z〉 →
(p(2nd ∘ 1st )b ↔ y =
b)) |
55 | 54 | exlimiv 1634 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (∃z p = 〈〈a, y〉, z〉 →
(p(2nd ∘ 1st )b ↔ y =
b)) |
56 | 49, 55 | sylbi 187 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (p1st 〈a, y〉 →
(p(2nd ∘ 1st )b ↔ y =
b)) |
57 | 56 | pm5.32i 618 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((p1st 〈a, y〉 ∧ p(2nd ∘ 1st )b) ↔ (p1st 〈a, y〉 ∧ y = b)) |
58 | 57 | exbii 1582 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (∃y(p1st 〈a, y〉 ∧ p(2nd ∘ 1st )b) ↔ ∃y(p1st 〈a, y〉 ∧ y = b)) |
59 | | exancom 1586 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (∃y(p1st 〈a, y〉 ∧ y = b) ↔ ∃y(y = b ∧ p1st 〈a, y〉)) |
60 | 58, 59 | bitri 240 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (∃y(p1st 〈a, y〉 ∧ p(2nd ∘ 1st )b) ↔ ∃y(y = b ∧ p1st 〈a, y〉)) |
61 | 47, 48, 60 | 3bitr2i 264 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((p(1st ∘ 1st )a ∧ p(2nd ∘ 1st )b) ↔ ∃y(y = b ∧ p1st 〈a, y〉)) |
62 | | opeq2 4579 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (y = b →
〈a,
y〉 =
〈a,
b〉) |
63 | 62 | breq2d 4651 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (y = b →
(p1st 〈a, y〉 ↔ p1st 〈a, b〉)) |
64 | 7, 63 | ceqsexv 2894 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (∃y(y = b ∧ p1st 〈a, y〉) ↔
p1st 〈a, b〉) |
65 | 61, 64 | bitri 240 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((p(1st ∘ 1st )a ∧ p(2nd ∘ 1st )b) ↔ p1st 〈a, b〉) |
66 | 65 | anbi1i 676 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((p(1st ∘ 1st )a ∧ p(2nd ∘ 1st )b) ∧ p2nd z) ↔ (p1st 〈a, b〉 ∧ p2nd z)) |
67 | 16, 7 | opex 4588 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 〈a, b〉 ∈ V |
68 | 67, 20 | op1st2nd 5790 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((p1st 〈a, b〉 ∧ p2nd z) ↔ p =
〈〈a, b〉, z〉) |
69 | 31, 66, 68 | 3bitri 262 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))〈a, 〈b, z〉〉 ↔ p =
〈〈a, b〉, z〉) |
70 | 69 | rexbii 2639 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∃p ∈ Cup p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))〈a, 〈b, z〉〉 ↔ ∃p ∈ Cup p = 〈〈a, b〉, z〉) |
71 | | elima 4754 |
. . . . . . . . . . . . . . . . . . 19
⊢ (〈a, 〈b, z〉〉 ∈
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ) ↔
∃p ∈ Cup p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))〈a, 〈b, z〉〉) |
72 | | risset 2661 |
. . . . . . . . . . . . . . . . . . 19
⊢ (〈〈a, b〉, z〉 ∈ Cup ↔ ∃p ∈ Cup p = 〈〈a, b〉, z〉) |
73 | 70, 71, 72 | 3bitr4i 268 |
. . . . . . . . . . . . . . . . . 18
⊢ (〈a, 〈b, z〉〉 ∈
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ) ↔
〈〈a, b〉, z〉 ∈ Cup ) |
74 | | df-br 4640 |
. . . . . . . . . . . . . . . . . 18
⊢ (〈a, b〉 Cup z ↔ 〈〈a, b〉, z〉 ∈ Cup ) |
75 | 16, 7 | brcup 5815 |
. . . . . . . . . . . . . . . . . 18
⊢ (〈a, b〉 Cup z ↔
z = (a
∪ b)) |
76 | 73, 74, 75 | 3bitr2i 264 |
. . . . . . . . . . . . . . . . 17
⊢ (〈a, 〈b, z〉〉 ∈
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ) ↔
z = (a
∪ b)) |
77 | 26, 76 | anbi12i 678 |
. . . . . . . . . . . . . . . 16
⊢ ((〈a, 〈b, z〉〉 ∈ Ins3 Disj ∧ 〈a, 〈b, z〉〉 ∈ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))
“ Cup )) ↔ ((a ∩ b) =
∅ ∧
z = (a
∪ b))) |
78 | 22, 77 | bitri 240 |
. . . . . . . . . . . . . . 15
⊢ (〈a, 〈b, z〉〉 ∈ ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup )) ↔
((a ∩ b) = ∅ ∧ z = (a ∪ b))) |
79 | 19, 21, 78 | 3bitri 262 |
. . . . . . . . . . . . . 14
⊢ (〈{a}, 〈{b}, 〈{z}, x〉〉〉 ∈ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup )) ↔
((a ∩ b) = ∅ ∧ z = (a ∪ b))) |
80 | 18, 79 | anbi12i 678 |
. . . . . . . . . . . . 13
⊢ ((〈{a}, 〈{b}, 〈{z}, x〉〉〉 ∈ Ins2 Ins2 S ∧ 〈{a}, 〈{b}, 〈{z}, x〉〉〉 ∈ Ins4 SI3
( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) ↔
(a ∈
x ∧
((a ∩ b) = ∅ ∧ z = (a ∪ b)))) |
81 | 12, 80 | bitri 240 |
. . . . . . . . . . . 12
⊢ (〈{a}, 〈{b}, 〈{z}, x〉〉〉 ∈ ( Ins2 Ins2 S ∩ Ins4 SI3
( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) ↔
(a ∈
x ∧
((a ∩ b) = ∅ ∧ z = (a ∪ b)))) |
82 | 81 | exbii 1582 |
. . . . . . . . . . 11
⊢ (∃a〈{a}, 〈{b}, 〈{z}, x〉〉〉 ∈ ( Ins2 Ins2 S ∩ Ins4 SI3
( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) ↔
∃a(a ∈ x ∧ ((a ∩
b) = ∅
∧ z =
(a ∪ b)))) |
83 | | elima1c 4947 |
. . . . . . . . . . 11
⊢ (〈{b}, 〈{z}, x〉〉 ∈ (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c) ↔ ∃a〈{a}, 〈{b}, 〈{z}, x〉〉〉 ∈ ( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup
)))) |
84 | | df-rex 2620 |
. . . . . . . . . . 11
⊢ (∃a ∈ x ((a ∩ b) =
∅ ∧
z = (a
∪ b)) ↔ ∃a(a ∈ x ∧ ((a ∩ b) =
∅ ∧
z = (a
∪ b)))) |
85 | 82, 83, 84 | 3bitr4i 268 |
. . . . . . . . . 10
⊢ (〈{b}, 〈{z}, x〉〉 ∈ (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c) ↔ ∃a ∈ x ((a ∩
b) = ∅
∧ z =
(a ∪ b))) |
86 | 11, 85 | bitri 240 |
. . . . . . . . 9
⊢ (〈{b}, 〈{z}, 〈x, y〉〉〉 ∈ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c) ↔ ∃a ∈ x ((a ∩
b) = ∅
∧ z =
(a ∪ b))) |
87 | 10, 86 | anbi12i 678 |
. . . . . . . 8
⊢ ((〈{b}, 〈{z}, 〈x, y〉〉〉 ∈ Ins2 Ins2 S ∧ 〈{b}, 〈{z}, 〈x, y〉〉〉 ∈ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c)) ↔ (b ∈ y ∧ ∃a ∈ x ((a ∩
b) = ∅
∧ z =
(a ∪ b)))) |
88 | 2, 87 | bitri 240 |
. . . . . . 7
⊢ (〈{b}, 〈{z}, 〈x, y〉〉〉 ∈ ( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c)) ↔ (b ∈ y ∧ ∃a ∈ x ((a ∩
b) = ∅
∧ z =
(a ∪ b)))) |
89 | 88 | exbii 1582 |
. . . . . 6
⊢ (∃b〈{b}, 〈{z}, 〈x, y〉〉〉 ∈ ( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c)) ↔ ∃b(b ∈ y ∧ ∃a ∈ x ((a ∩
b) = ∅
∧ z =
(a ∪ b)))) |
90 | | elima1c 4947 |
. . . . . 6
⊢ (〈{z}, 〈x, y〉〉 ∈ (( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c)) “ 1c) ↔ ∃b〈{b}, 〈{z}, 〈x, y〉〉〉 ∈ ( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c))) |
91 | | df-rex 2620 |
. . . . . 6
⊢ (∃b ∈ y ∃a ∈ x ((a ∩ b) =
∅ ∧
z = (a
∪ b)) ↔ ∃b(b ∈ y ∧ ∃a ∈ x ((a ∩ b) =
∅ ∧
z = (a
∪ b)))) |
92 | 89, 90, 91 | 3bitr4i 268 |
. . . . 5
⊢ (〈{z}, 〈x, y〉〉 ∈ (( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c)) “ 1c) ↔ ∃b ∈ y ∃a ∈ x ((a ∩ b) =
∅ ∧
z = (a
∪ b))) |
93 | | eladdc 4398 |
. . . . . 6
⊢ (z ∈ (x +c y) ↔ ∃a ∈ x ∃b ∈ y ((a ∩ b) =
∅ ∧
z = (a
∪ b))) |
94 | | rexcom 2772 |
. . . . . 6
⊢ (∃a ∈ x ∃b ∈ y ((a ∩ b) =
∅ ∧
z = (a
∪ b)) ↔ ∃b ∈ y ∃a ∈ x ((a ∩ b) =
∅ ∧
z = (a
∪ b))) |
95 | 93, 94 | bitri 240 |
. . . . 5
⊢ (z ∈ (x +c y) ↔ ∃b ∈ y ∃a ∈ x ((a ∩ b) =
∅ ∧
z = (a
∪ b))) |
96 | 92, 95 | bitr4i 243 |
. . . 4
⊢ (〈{z}, 〈x, y〉〉 ∈ (( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c)) “ 1c) ↔ z ∈ (x +c y)) |
97 | 96 | releqmpt2 5809 |
. . 3
⊢ (((V × V)
× V) ∖ (( Ins2 S ⊕ Ins3 (( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c)) “ 1c)) “
1c)) = (x ∈ V, y ∈ V ↦ (x +c y)) |
98 | 1, 97 | eqtr4i 2376 |
. 2
⊢ AddC = (((V × V) × V) ∖ (( Ins2 S ⊕ Ins3 (( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c)) “ 1c)) “
1c)) |
99 | | vvex 4109 |
. . 3
⊢ V ∈ V |
100 | | ssetex 4744 |
. . . . . . 7
⊢ S ∈
V |
101 | 100 | ins2ex 5797 |
. . . . . 6
⊢ Ins2 S ∈ V |
102 | 101 | ins2ex 5797 |
. . . . 5
⊢ Ins2 Ins2 S ∈
V |
103 | | disjex 5823 |
. . . . . . . . . . . 12
⊢ Disj ∈
V |
104 | 103 | ins3ex 5798 |
. . . . . . . . . . 11
⊢ Ins3 Disj ∈ V |
105 | | 1stex 4739 |
. . . . . . . . . . . . . 14
⊢ 1st
∈ V |
106 | 105, 105 | coex 4750 |
. . . . . . . . . . . . 13
⊢ (1st
∘ 1st ) ∈ V |
107 | | 2ndex 5112 |
. . . . . . . . . . . . . . 15
⊢ 2nd
∈ V |
108 | 107, 105 | coex 4750 |
. . . . . . . . . . . . . 14
⊢ (2nd
∘ 1st ) ∈ V |
109 | 108, 107 | txpex 5785 |
. . . . . . . . . . . . 13
⊢ ((2nd
∘ 1st ) ⊗ 2nd )
∈ V |
110 | 106, 109 | txpex 5785 |
. . . . . . . . . . . 12
⊢ ((1st
∘ 1st ) ⊗ ((2nd
∘ 1st ) ⊗ 2nd
)) ∈ V |
111 | | cupex 5816 |
. . . . . . . . . . . 12
⊢ Cup ∈
V |
112 | 110, 111 | imaex 4747 |
. . . . . . . . . . 11
⊢ (((1st
∘ 1st ) ⊗ ((2nd
∘ 1st ) ⊗ 2nd
)) “ Cup ) ∈ V |
113 | 104, 112 | inex 4105 |
. . . . . . . . . 10
⊢ ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup )) ∈ V |
114 | 113 | si3ex 5806 |
. . . . . . . . 9
⊢ SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup )) ∈ V |
115 | 114 | ins4ex 5799 |
. . . . . . . 8
⊢ Ins4 SI3
( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup )) ∈ V |
116 | 102, 115 | inex 4105 |
. . . . . . 7
⊢ ( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) ∈ V |
117 | | 1cex 4142 |
. . . . . . 7
⊢
1c ∈
V |
118 | 116, 117 | imaex 4747 |
. . . . . 6
⊢ (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c) ∈ V |
119 | 118 | ins4ex 5799 |
. . . . 5
⊢ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c) ∈ V |
120 | 102, 119 | inex 4105 |
. . . 4
⊢ ( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c)) ∈ V |
121 | 120, 117 | imaex 4747 |
. . 3
⊢ (( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c)) “ 1c) ∈ V |
122 | 99, 99, 121 | mpt2exlem 5811 |
. 2
⊢ (((V × V)
× V) ∖ (( Ins2 S ⊕ Ins3 (( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins3 Disj ∩
(((1st ∘ 1st )
⊗ ((2nd ∘ 1st )
⊗ 2nd )) “ Cup ))) “
1c)) “ 1c)) “
1c)) ∈ V |
123 | 98, 122 | eqeltri 2423 |
1
⊢ AddC ∈
V |