Proof of Theorem uneqdifeq
| Step | Hyp | Ref
| Expression |
| 1 | | uncom 4158 |
. . . . 5
⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) |
| 2 | | eqtr 2760 |
. . . . . . 7
⊢ (((𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) = 𝐶) → (𝐵 ∪ 𝐴) = 𝐶) |
| 3 | 2 | eqcomd 2743 |
. . . . . 6
⊢ (((𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) = 𝐶) → 𝐶 = (𝐵 ∪ 𝐴)) |
| 4 | | difeq1 4119 |
. . . . . . 7
⊢ (𝐶 = (𝐵 ∪ 𝐴) → (𝐶 ∖ 𝐴) = ((𝐵 ∪ 𝐴) ∖ 𝐴)) |
| 5 | | difun2 4481 |
. . . . . . 7
⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴) |
| 6 | | eqtr 2760 |
. . . . . . . 8
⊢ (((𝐶 ∖ 𝐴) = ((𝐵 ∪ 𝐴) ∖ 𝐴) ∧ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)) → (𝐶 ∖ 𝐴) = (𝐵 ∖ 𝐴)) |
| 7 | | incom 4209 |
. . . . . . . . . . 11
⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) |
| 8 | 7 | eqeq1i 2742 |
. . . . . . . . . 10
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅) |
| 9 | | disj3 4454 |
. . . . . . . . . 10
⊢ ((𝐵 ∩ 𝐴) = ∅ ↔ 𝐵 = (𝐵 ∖ 𝐴)) |
| 10 | 8, 9 | bitri 275 |
. . . . . . . . 9
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐵 = (𝐵 ∖ 𝐴)) |
| 11 | | eqtr 2760 |
. . . . . . . . . . 11
⊢ (((𝐶 ∖ 𝐴) = (𝐵 ∖ 𝐴) ∧ (𝐵 ∖ 𝐴) = 𝐵) → (𝐶 ∖ 𝐴) = 𝐵) |
| 12 | 11 | expcom 413 |
. . . . . . . . . 10
⊢ ((𝐵 ∖ 𝐴) = 𝐵 → ((𝐶 ∖ 𝐴) = (𝐵 ∖ 𝐴) → (𝐶 ∖ 𝐴) = 𝐵)) |
| 13 | 12 | eqcoms 2745 |
. . . . . . . . 9
⊢ (𝐵 = (𝐵 ∖ 𝐴) → ((𝐶 ∖ 𝐴) = (𝐵 ∖ 𝐴) → (𝐶 ∖ 𝐴) = 𝐵)) |
| 14 | 10, 13 | sylbi 217 |
. . . . . . . 8
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 ∖ 𝐴) = (𝐵 ∖ 𝐴) → (𝐶 ∖ 𝐴) = 𝐵)) |
| 15 | 6, 14 | syl5com 31 |
. . . . . . 7
⊢ (((𝐶 ∖ 𝐴) = ((𝐵 ∪ 𝐴) ∖ 𝐴) ∧ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)) → ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ∖ 𝐴) = 𝐵)) |
| 16 | 4, 5, 15 | sylancl 586 |
. . . . . 6
⊢ (𝐶 = (𝐵 ∪ 𝐴) → ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ∖ 𝐴) = 𝐵)) |
| 17 | 3, 16 | syl 17 |
. . . . 5
⊢ (((𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) = 𝐶) → ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ∖ 𝐴) = 𝐵)) |
| 18 | 1, 17 | mpan 690 |
. . . 4
⊢ ((𝐴 ∪ 𝐵) = 𝐶 → ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ∖ 𝐴) = 𝐵)) |
| 19 | 18 | com12 32 |
. . 3
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) = 𝐶 → (𝐶 ∖ 𝐴) = 𝐵)) |
| 20 | 19 | adantl 481 |
. 2
⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝐶 → (𝐶 ∖ 𝐴) = 𝐵)) |
| 21 | | simpl 482 |
. . . . . 6
⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐶 ∖ 𝐴) = 𝐵) → 𝐴 ⊆ 𝐶) |
| 22 | | difssd 4137 |
. . . . . . . 8
⊢ ((𝐶 ∖ 𝐴) = 𝐵 → (𝐶 ∖ 𝐴) ⊆ 𝐶) |
| 23 | | sseq1 4009 |
. . . . . . . 8
⊢ ((𝐶 ∖ 𝐴) = 𝐵 → ((𝐶 ∖ 𝐴) ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
| 24 | 22, 23 | mpbid 232 |
. . . . . . 7
⊢ ((𝐶 ∖ 𝐴) = 𝐵 → 𝐵 ⊆ 𝐶) |
| 25 | 24 | adantl 481 |
. . . . . 6
⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐶 ∖ 𝐴) = 𝐵) → 𝐵 ⊆ 𝐶) |
| 26 | 21, 25 | unssd 4192 |
. . . . 5
⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐶 ∖ 𝐴) = 𝐵) → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
| 27 | | eqimss 4042 |
. . . . . . 7
⊢ ((𝐶 ∖ 𝐴) = 𝐵 → (𝐶 ∖ 𝐴) ⊆ 𝐵) |
| 28 | | ssundif 4488 |
. . . . . . 7
⊢ (𝐶 ⊆ (𝐴 ∪ 𝐵) ↔ (𝐶 ∖ 𝐴) ⊆ 𝐵) |
| 29 | 27, 28 | sylibr 234 |
. . . . . 6
⊢ ((𝐶 ∖ 𝐴) = 𝐵 → 𝐶 ⊆ (𝐴 ∪ 𝐵)) |
| 30 | 29 | adantl 481 |
. . . . 5
⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐶 ∖ 𝐴) = 𝐵) → 𝐶 ⊆ (𝐴 ∪ 𝐵)) |
| 31 | 26, 30 | eqssd 4001 |
. . . 4
⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐶 ∖ 𝐴) = 𝐵) → (𝐴 ∪ 𝐵) = 𝐶) |
| 32 | 31 | ex 412 |
. . 3
⊢ (𝐴 ⊆ 𝐶 → ((𝐶 ∖ 𝐴) = 𝐵 → (𝐴 ∪ 𝐵) = 𝐶)) |
| 33 | 32 | adantr 480 |
. 2
⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐶 ∖ 𝐴) = 𝐵 → (𝐴 ∪ 𝐵) = 𝐶)) |
| 34 | 20, 33 | impbid 212 |
1
⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝐶 ↔ (𝐶 ∖ 𝐴) = 𝐵)) |