| Step | Hyp | Ref
| Expression |
| 1 | | eleq2 2830 |
. . . . 5
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) →
(〈𝑥, ∅〉
∈ ((𝐴 ×
{∅}) ∪ (𝐵 ×
{{∅}})) ↔ 〈𝑥, ∅〉 ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))) |
| 2 | | 0ex 5307 |
. . . . . . . . 9
⊢ ∅
∈ V |
| 3 | 2 | snid 4662 |
. . . . . . . 8
⊢ ∅
∈ {∅} |
| 4 | | opelxp 5721 |
. . . . . . . 8
⊢
(〈𝑥,
∅〉 ∈ (𝐴
× {∅}) ↔ (𝑥 ∈ 𝐴 ∧ ∅ ∈
{∅})) |
| 5 | 3, 4 | mpbiran2 710 |
. . . . . . 7
⊢
(〈𝑥,
∅〉 ∈ (𝐴
× {∅}) ↔ 𝑥
∈ 𝐴) |
| 6 | | opelxp 5721 |
. . . . . . . 8
⊢
(〈𝑥,
∅〉 ∈ (𝐵
× {{∅}}) ↔ (𝑥 ∈ 𝐵 ∧ ∅ ∈
{{∅}})) |
| 7 | | 0nep0 5358 |
. . . . . . . . . 10
⊢ ∅
≠ {∅} |
| 8 | 2 | elsn 4641 |
. . . . . . . . . 10
⊢ (∅
∈ {{∅}} ↔ ∅ = {∅}) |
| 9 | 7, 8 | nemtbir 3038 |
. . . . . . . . 9
⊢ ¬
∅ ∈ {{∅}} |
| 10 | 9 | bianfi 533 |
. . . . . . . 8
⊢ (∅
∈ {{∅}} ↔ (𝑥 ∈ 𝐵 ∧ ∅ ∈
{{∅}})) |
| 11 | 6, 10 | bitr4i 278 |
. . . . . . 7
⊢
(〈𝑥,
∅〉 ∈ (𝐵
× {{∅}}) ↔ ∅ ∈ {{∅}}) |
| 12 | 5, 11 | orbi12i 915 |
. . . . . 6
⊢
((〈𝑥,
∅〉 ∈ (𝐴
× {∅}) ∨ 〈𝑥, ∅〉 ∈ (𝐵 × {{∅}})) ↔ (𝑥 ∈ 𝐴 ∨ ∅ ∈
{{∅}})) |
| 13 | | elun 4153 |
. . . . . 6
⊢
(〈𝑥,
∅〉 ∈ ((𝐴
× {∅}) ∪ (𝐵
× {{∅}})) ↔ (〈𝑥, ∅〉 ∈ (𝐴 × {∅}) ∨ 〈𝑥, ∅〉 ∈ (𝐵 ×
{{∅}}))) |
| 14 | 9 | biorfri 940 |
. . . . . 6
⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ ∅ ∈
{{∅}})) |
| 15 | 12, 13, 14 | 3bitr4ri 304 |
. . . . 5
⊢ (𝑥 ∈ 𝐴 ↔ 〈𝑥, ∅〉 ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}}))) |
| 16 | | opelxp 5721 |
. . . . . . . 8
⊢
(〈𝑥,
∅〉 ∈ (𝐶
× {∅}) ↔ (𝑥 ∈ 𝐶 ∧ ∅ ∈
{∅})) |
| 17 | 3, 16 | mpbiran2 710 |
. . . . . . 7
⊢
(〈𝑥,
∅〉 ∈ (𝐶
× {∅}) ↔ 𝑥
∈ 𝐶) |
| 18 | | opelxp 5721 |
. . . . . . . 8
⊢
(〈𝑥,
∅〉 ∈ (𝐷
× {{∅}}) ↔ (𝑥 ∈ 𝐷 ∧ ∅ ∈
{{∅}})) |
| 19 | 9 | bianfi 533 |
. . . . . . . 8
⊢ (∅
∈ {{∅}} ↔ (𝑥 ∈ 𝐷 ∧ ∅ ∈
{{∅}})) |
| 20 | 18, 19 | bitr4i 278 |
. . . . . . 7
⊢
(〈𝑥,
∅〉 ∈ (𝐷
× {{∅}}) ↔ ∅ ∈ {{∅}}) |
| 21 | 17, 20 | orbi12i 915 |
. . . . . 6
⊢
((〈𝑥,
∅〉 ∈ (𝐶
× {∅}) ∨ 〈𝑥, ∅〉 ∈ (𝐷 × {{∅}})) ↔ (𝑥 ∈ 𝐶 ∨ ∅ ∈
{{∅}})) |
| 22 | | elun 4153 |
. . . . . 6
⊢
(〈𝑥,
∅〉 ∈ ((𝐶
× {∅}) ∪ (𝐷
× {{∅}})) ↔ (〈𝑥, ∅〉 ∈ (𝐶 × {∅}) ∨ 〈𝑥, ∅〉 ∈ (𝐷 ×
{{∅}}))) |
| 23 | 9 | biorfri 940 |
. . . . . 6
⊢ (𝑥 ∈ 𝐶 ↔ (𝑥 ∈ 𝐶 ∨ ∅ ∈
{{∅}})) |
| 24 | 21, 22, 23 | 3bitr4ri 304 |
. . . . 5
⊢ (𝑥 ∈ 𝐶 ↔ 〈𝑥, ∅〉 ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))) |
| 25 | 1, 15, 24 | 3bitr4g 314 |
. . . 4
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) →
(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶)) |
| 26 | 25 | eqrdv 2735 |
. . 3
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) →
𝐴 = 𝐶) |
| 27 | | eleq2 2830 |
. . . . 5
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) →
(〈𝑥, {∅}〉
∈ ((𝐴 ×
{∅}) ∪ (𝐵 ×
{{∅}})) ↔ 〈𝑥, {∅}〉 ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})))) |
| 28 | | opelxp 5721 |
. . . . . . . 8
⊢
(〈𝑥,
{∅}〉 ∈ (𝐴
× {∅}) ↔ (𝑥 ∈ 𝐴 ∧ {∅} ∈
{∅})) |
| 29 | | snex 5436 |
. . . . . . . . . . . 12
⊢ {∅}
∈ V |
| 30 | 29 | elsn 4641 |
. . . . . . . . . . 11
⊢
({∅} ∈ {∅} ↔ {∅} = ∅) |
| 31 | | eqcom 2744 |
. . . . . . . . . . 11
⊢
({∅} = ∅ ↔ ∅ = {∅}) |
| 32 | 30, 31 | bitri 275 |
. . . . . . . . . 10
⊢
({∅} ∈ {∅} ↔ ∅ = {∅}) |
| 33 | 7, 32 | nemtbir 3038 |
. . . . . . . . 9
⊢ ¬
{∅} ∈ {∅} |
| 34 | 33 | bianfi 533 |
. . . . . . . 8
⊢
({∅} ∈ {∅} ↔ (𝑥 ∈ 𝐴 ∧ {∅} ∈
{∅})) |
| 35 | 28, 34 | bitr4i 278 |
. . . . . . 7
⊢
(〈𝑥,
{∅}〉 ∈ (𝐴
× {∅}) ↔ {∅} ∈ {∅}) |
| 36 | 29 | snid 4662 |
. . . . . . . 8
⊢ {∅}
∈ {{∅}} |
| 37 | | opelxp 5721 |
. . . . . . . 8
⊢
(〈𝑥,
{∅}〉 ∈ (𝐵
× {{∅}}) ↔ (𝑥 ∈ 𝐵 ∧ {∅} ∈
{{∅}})) |
| 38 | 36, 37 | mpbiran2 710 |
. . . . . . 7
⊢
(〈𝑥,
{∅}〉 ∈ (𝐵
× {{∅}}) ↔ 𝑥 ∈ 𝐵) |
| 39 | 35, 38 | orbi12i 915 |
. . . . . 6
⊢
((〈𝑥,
{∅}〉 ∈ (𝐴
× {∅}) ∨ 〈𝑥, {∅}〉 ∈ (𝐵 × {{∅}})) ↔ ({∅}
∈ {∅} ∨ 𝑥
∈ 𝐵)) |
| 40 | | elun 4153 |
. . . . . 6
⊢
(〈𝑥,
{∅}〉 ∈ ((𝐴
× {∅}) ∪ (𝐵
× {{∅}})) ↔ (〈𝑥, {∅}〉 ∈ (𝐴 × {∅}) ∨ 〈𝑥, {∅}〉 ∈ (𝐵 ×
{{∅}}))) |
| 41 | 33 | biorfi 939 |
. . . . . 6
⊢ (𝑥 ∈ 𝐵 ↔ ({∅} ∈ {∅} ∨
𝑥 ∈ 𝐵)) |
| 42 | 39, 40, 41 | 3bitr4ri 304 |
. . . . 5
⊢ (𝑥 ∈ 𝐵 ↔ 〈𝑥, {∅}〉 ∈ ((𝐴 × {∅}) ∪ (𝐵 × {{∅}}))) |
| 43 | | opelxp 5721 |
. . . . . . . 8
⊢
(〈𝑥,
{∅}〉 ∈ (𝐶
× {∅}) ↔ (𝑥 ∈ 𝐶 ∧ {∅} ∈
{∅})) |
| 44 | 33 | bianfi 533 |
. . . . . . . 8
⊢
({∅} ∈ {∅} ↔ (𝑥 ∈ 𝐶 ∧ {∅} ∈
{∅})) |
| 45 | 43, 44 | bitr4i 278 |
. . . . . . 7
⊢
(〈𝑥,
{∅}〉 ∈ (𝐶
× {∅}) ↔ {∅} ∈ {∅}) |
| 46 | | opelxp 5721 |
. . . . . . . 8
⊢
(〈𝑥,
{∅}〉 ∈ (𝐷
× {{∅}}) ↔ (𝑥 ∈ 𝐷 ∧ {∅} ∈
{{∅}})) |
| 47 | 36, 46 | mpbiran2 710 |
. . . . . . 7
⊢
(〈𝑥,
{∅}〉 ∈ (𝐷
× {{∅}}) ↔ 𝑥 ∈ 𝐷) |
| 48 | 45, 47 | orbi12i 915 |
. . . . . 6
⊢
((〈𝑥,
{∅}〉 ∈ (𝐶
× {∅}) ∨ 〈𝑥, {∅}〉 ∈ (𝐷 × {{∅}})) ↔ ({∅}
∈ {∅} ∨ 𝑥
∈ 𝐷)) |
| 49 | | elun 4153 |
. . . . . 6
⊢
(〈𝑥,
{∅}〉 ∈ ((𝐶
× {∅}) ∪ (𝐷
× {{∅}})) ↔ (〈𝑥, {∅}〉 ∈ (𝐶 × {∅}) ∨ 〈𝑥, {∅}〉 ∈ (𝐷 ×
{{∅}}))) |
| 50 | 33 | biorfi 939 |
. . . . . 6
⊢ (𝑥 ∈ 𝐷 ↔ ({∅} ∈ {∅} ∨
𝑥 ∈ 𝐷)) |
| 51 | 48, 49, 50 | 3bitr4ri 304 |
. . . . 5
⊢ (𝑥 ∈ 𝐷 ↔ 〈𝑥, {∅}〉 ∈ ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))) |
| 52 | 27, 42, 51 | 3bitr4g 314 |
. . . 4
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) →
(𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐷)) |
| 53 | 52 | eqrdv 2735 |
. . 3
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) →
𝐵 = 𝐷) |
| 54 | 26, 53 | jca 511 |
. 2
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) →
(𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
| 55 | | xpeq1 5699 |
. . 3
⊢ (𝐴 = 𝐶 → (𝐴 × {∅}) = (𝐶 × {∅})) |
| 56 | | xpeq1 5699 |
. . 3
⊢ (𝐵 = 𝐷 → (𝐵 × {{∅}}) = (𝐷 × {{∅}})) |
| 57 | | uneq12 4163 |
. . 3
⊢ (((𝐴 × {∅}) = (𝐶 × {∅}) ∧ (𝐵 × {{∅}}) = (𝐷 × {{∅}})) →
((𝐴 × {∅})
∪ (𝐵 ×
{{∅}})) = ((𝐶 ×
{∅}) ∪ (𝐷 ×
{{∅}}))) |
| 58 | 55, 56, 57 | syl2an 596 |
. 2
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}}))) |
| 59 | 54, 58 | impbii 209 |
1
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔
(𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |