Step | Hyp | Ref
| Expression |
1 | | eleq2 2221 |
. . . . 5
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) →
(〈𝑥, ∅〉
∈ ((𝐴 ×
{∅}) ∪ (𝐵 ×
{{∅}})) ↔ 〈𝑥, ∅〉 ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))) |
2 | | 0ex 4091 |
. . . . . . . . 9
⊢ ∅
∈ V |
3 | 2 | snid 3591 |
. . . . . . . 8
⊢ ∅
∈ {∅} |
4 | | opelxp 4616 |
. . . . . . . 8
⊢
(〈𝑥,
∅〉 ∈ (𝐴
× {∅}) ↔ (𝑥 ∈ 𝐴 ∧ ∅ ∈
{∅})) |
5 | 3, 4 | mpbiran2 926 |
. . . . . . 7
⊢
(〈𝑥,
∅〉 ∈ (𝐴
× {∅}) ↔ 𝑥
∈ 𝐴) |
6 | | opelxp 4616 |
. . . . . . . 8
⊢
(〈𝑥,
∅〉 ∈ (𝐵
× {{∅}}) ↔ (𝑥 ∈ 𝐵 ∧ ∅ ∈
{{∅}})) |
7 | | 0nep0 4126 |
. . . . . . . . . 10
⊢ ∅
≠ {∅} |
8 | 2 | elsn 3576 |
. . . . . . . . . 10
⊢ (∅
∈ {{∅}} ↔ ∅ = {∅}) |
9 | 7, 8 | nemtbir 2416 |
. . . . . . . . 9
⊢ ¬
∅ ∈ {{∅}} |
10 | 9 | bianfi 932 |
. . . . . . . 8
⊢ (∅
∈ {{∅}} ↔ (𝑥 ∈ 𝐵 ∧ ∅ ∈
{{∅}})) |
11 | 6, 10 | bitr4i 186 |
. . . . . . 7
⊢
(〈𝑥,
∅〉 ∈ (𝐵
× {{∅}}) ↔ ∅ ∈ {{∅}}) |
12 | 5, 11 | orbi12i 754 |
. . . . . 6
⊢
((〈𝑥,
∅〉 ∈ (𝐴
× {∅}) ∨ 〈𝑥, ∅〉 ∈ (𝐵 × {{∅}})) ↔ (𝑥 ∈ 𝐴 ∨ ∅ ∈
{{∅}})) |
13 | | elun 3248 |
. . . . . 6
⊢
(〈𝑥,
∅〉 ∈ ((𝐴
× {∅}) ∪ (𝐵
× {{∅}})) ↔ (〈𝑥, ∅〉 ∈ (𝐴 × {∅}) ∨ 〈𝑥, ∅〉 ∈ (𝐵 ×
{{∅}}))) |
14 | 9 | biorfi 736 |
. . . . . 6
⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ ∅ ∈
{{∅}})) |
15 | 12, 13, 14 | 3bitr4ri 212 |
. . . . 5
⊢ (𝑥 ∈ 𝐴 ↔ 〈𝑥, ∅〉 ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}}))) |
16 | | opelxp 4616 |
. . . . . . . 8
⊢
(〈𝑥,
∅〉 ∈ (𝐶
× {∅}) ↔ (𝑥 ∈ 𝐶 ∧ ∅ ∈
{∅})) |
17 | 3, 16 | mpbiran2 926 |
. . . . . . 7
⊢
(〈𝑥,
∅〉 ∈ (𝐶
× {∅}) ↔ 𝑥
∈ 𝐶) |
18 | | opelxp 4616 |
. . . . . . . 8
⊢
(〈𝑥,
∅〉 ∈ (𝐷
× {{∅}}) ↔ (𝑥 ∈ 𝐷 ∧ ∅ ∈
{{∅}})) |
19 | 9 | bianfi 932 |
. . . . . . . 8
⊢ (∅
∈ {{∅}} ↔ (𝑥 ∈ 𝐷 ∧ ∅ ∈
{{∅}})) |
20 | 18, 19 | bitr4i 186 |
. . . . . . 7
⊢
(〈𝑥,
∅〉 ∈ (𝐷
× {{∅}}) ↔ ∅ ∈ {{∅}}) |
21 | 17, 20 | orbi12i 754 |
. . . . . 6
⊢
((〈𝑥,
∅〉 ∈ (𝐶
× {∅}) ∨ 〈𝑥, ∅〉 ∈ (𝐷 × {{∅}})) ↔ (𝑥 ∈ 𝐶 ∨ ∅ ∈
{{∅}})) |
22 | | elun 3248 |
. . . . . 6
⊢
(〈𝑥,
∅〉 ∈ ((𝐶
× {∅}) ∪ (𝐷
× {{∅}})) ↔ (〈𝑥, ∅〉 ∈ (𝐶 × {∅}) ∨ 〈𝑥, ∅〉 ∈ (𝐷 ×
{{∅}}))) |
23 | 9 | biorfi 736 |
. . . . . 6
⊢ (𝑥 ∈ 𝐶 ↔ (𝑥 ∈ 𝐶 ∨ ∅ ∈
{{∅}})) |
24 | 21, 22, 23 | 3bitr4ri 212 |
. . . . 5
⊢ (𝑥 ∈ 𝐶 ↔ 〈𝑥, ∅〉 ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))) |
25 | 1, 15, 24 | 3bitr4g 222 |
. . . 4
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) →
(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶)) |
26 | 25 | eqrdv 2155 |
. . 3
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) →
𝐴 = 𝐶) |
27 | | eleq2 2221 |
. . . . 5
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) →
(〈𝑥, {∅}〉
∈ ((𝐴 ×
{∅}) ∪ (𝐵 ×
{{∅}})) ↔ 〈𝑥, {∅}〉 ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))) |
28 | | opelxp 4616 |
. . . . . . . 8
⊢
(〈𝑥,
{∅}〉 ∈ (𝐴
× {∅}) ↔ (𝑥 ∈ 𝐴 ∧ {∅} ∈
{∅})) |
29 | | p0ex 4149 |
. . . . . . . . . . . 12
⊢ {∅}
∈ V |
30 | 29 | elsn 3576 |
. . . . . . . . . . 11
⊢
({∅} ∈ {∅} ↔ {∅} = ∅) |
31 | | eqcom 2159 |
. . . . . . . . . . 11
⊢
({∅} = ∅ ↔ ∅ = {∅}) |
32 | 30, 31 | bitri 183 |
. . . . . . . . . 10
⊢
({∅} ∈ {∅} ↔ ∅ = {∅}) |
33 | 7, 32 | nemtbir 2416 |
. . . . . . . . 9
⊢ ¬
{∅} ∈ {∅} |
34 | 33 | bianfi 932 |
. . . . . . . 8
⊢
({∅} ∈ {∅} ↔ (𝑥 ∈ 𝐴 ∧ {∅} ∈
{∅})) |
35 | 28, 34 | bitr4i 186 |
. . . . . . 7
⊢
(〈𝑥,
{∅}〉 ∈ (𝐴
× {∅}) ↔ {∅} ∈ {∅}) |
36 | 29 | snid 3591 |
. . . . . . . 8
⊢ {∅}
∈ {{∅}} |
37 | | opelxp 4616 |
. . . . . . . 8
⊢
(〈𝑥,
{∅}〉 ∈ (𝐵
× {{∅}}) ↔ (𝑥 ∈ 𝐵 ∧ {∅} ∈
{{∅}})) |
38 | 36, 37 | mpbiran2 926 |
. . . . . . 7
⊢
(〈𝑥,
{∅}〉 ∈ (𝐵
× {{∅}}) ↔ 𝑥 ∈ 𝐵) |
39 | 35, 38 | orbi12i 754 |
. . . . . 6
⊢
((〈𝑥,
{∅}〉 ∈ (𝐴
× {∅}) ∨ 〈𝑥, {∅}〉 ∈ (𝐵 × {{∅}})) ↔ ({∅}
∈ {∅} ∨ 𝑥
∈ 𝐵)) |
40 | | elun 3248 |
. . . . . 6
⊢
(〈𝑥,
{∅}〉 ∈ ((𝐴
× {∅}) ∪ (𝐵
× {{∅}})) ↔ (〈𝑥, {∅}〉 ∈ (𝐴 × {∅}) ∨ 〈𝑥, {∅}〉 ∈ (𝐵 ×
{{∅}}))) |
41 | | biorf 734 |
. . . . . . 7
⊢ (¬
{∅} ∈ {∅} → (𝑥 ∈ 𝐵 ↔ ({∅} ∈ {∅} ∨
𝑥 ∈ 𝐵))) |
42 | 33, 41 | ax-mp 5 |
. . . . . 6
⊢ (𝑥 ∈ 𝐵 ↔ ({∅} ∈ {∅} ∨
𝑥 ∈ 𝐵)) |
43 | 39, 40, 42 | 3bitr4ri 212 |
. . . . 5
⊢ (𝑥 ∈ 𝐵 ↔ 〈𝑥, {∅}〉 ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}}))) |
44 | | opelxp 4616 |
. . . . . . . 8
⊢
(〈𝑥,
{∅}〉 ∈ (𝐶
× {∅}) ↔ (𝑥 ∈ 𝐶 ∧ {∅} ∈
{∅})) |
45 | 33 | bianfi 932 |
. . . . . . . 8
⊢
({∅} ∈ {∅} ↔ (𝑥 ∈ 𝐶 ∧ {∅} ∈
{∅})) |
46 | 44, 45 | bitr4i 186 |
. . . . . . 7
⊢
(〈𝑥,
{∅}〉 ∈ (𝐶
× {∅}) ↔ {∅} ∈ {∅}) |
47 | | opelxp 4616 |
. . . . . . . 8
⊢
(〈𝑥,
{∅}〉 ∈ (𝐷
× {{∅}}) ↔ (𝑥 ∈ 𝐷 ∧ {∅} ∈
{{∅}})) |
48 | 36, 47 | mpbiran2 926 |
. . . . . . 7
⊢
(〈𝑥,
{∅}〉 ∈ (𝐷
× {{∅}}) ↔ 𝑥 ∈ 𝐷) |
49 | 46, 48 | orbi12i 754 |
. . . . . 6
⊢
((〈𝑥,
{∅}〉 ∈ (𝐶
× {∅}) ∨ 〈𝑥, {∅}〉 ∈ (𝐷 × {{∅}})) ↔ ({∅}
∈ {∅} ∨ 𝑥
∈ 𝐷)) |
50 | | elun 3248 |
. . . . . 6
⊢
(〈𝑥,
{∅}〉 ∈ ((𝐶
× {∅}) ∪ (𝐷
× {{∅}})) ↔ (〈𝑥, {∅}〉 ∈ (𝐶 × {∅}) ∨ 〈𝑥, {∅}〉 ∈ (𝐷 ×
{{∅}}))) |
51 | | biorf 734 |
. . . . . . 7
⊢ (¬
{∅} ∈ {∅} → (𝑥 ∈ 𝐷 ↔ ({∅} ∈ {∅} ∨
𝑥 ∈ 𝐷))) |
52 | 33, 51 | ax-mp 5 |
. . . . . 6
⊢ (𝑥 ∈ 𝐷 ↔ ({∅} ∈ {∅} ∨
𝑥 ∈ 𝐷)) |
53 | 49, 50, 52 | 3bitr4ri 212 |
. . . . 5
⊢ (𝑥 ∈ 𝐷 ↔ 〈𝑥, {∅}〉 ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))) |
54 | 27, 43, 53 | 3bitr4g 222 |
. . . 4
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) →
(𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐷)) |
55 | 54 | eqrdv 2155 |
. . 3
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) →
𝐵 = 𝐷) |
56 | 26, 55 | jca 304 |
. 2
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) →
(𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
57 | | xpeq1 4600 |
. . 3
⊢ (𝐴 = 𝐶 → (𝐴 × {∅}) = (𝐶 × {∅})) |
58 | | xpeq1 4600 |
. . 3
⊢ (𝐵 = 𝐷 → (𝐵 × {{∅}}) = (𝐷 × {{∅}})) |
59 | | uneq12 3256 |
. . 3
⊢ (((𝐴 × {∅}) = (𝐶 × {∅}) ∧ (𝐵 × {{∅}}) = (𝐷 × {{∅}})) →
((𝐴 × {∅})
∪ (𝐵 ×
{{∅}})) = ((𝐶 ×
{∅}) ∪ (𝐷 ×
{{∅}}))) |
60 | 57, 58, 59 | syl2an 287 |
. 2
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))) |
61 | 56, 60 | impbii 125 |
1
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔
(𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |