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Mirrors > Home > MPE Home > Th. List > disjeq0 | Structured version Visualization version GIF version |
Description: Two disjoint sets are equal iff both are empty. (Contributed by AV, 19-Jun-2022.) |
Ref | Expression |
---|---|
disjeq0 | ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = 𝐵 ↔ (𝐴 = ∅ ∧ 𝐵 = ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1 4206 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐵)) | |
2 | inidm 4220 | . . . . . 6 ⊢ (𝐵 ∩ 𝐵) = 𝐵 | |
3 | 1, 2 | eqtrdi 2782 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐵) |
4 | 3 | eqeq1d 2728 | . . . 4 ⊢ (𝐴 = 𝐵 → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐵 = ∅)) |
5 | eqtr 2749 | . . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = ∅) → 𝐴 = ∅) | |
6 | simpr 483 | . . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = ∅) → 𝐵 = ∅) | |
7 | 5, 6 | jca 510 | . . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = ∅) → (𝐴 = ∅ ∧ 𝐵 = ∅)) |
8 | 7 | ex 411 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐵 = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅))) |
9 | 4, 8 | sylbid 239 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅))) |
10 | 9 | com12 32 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = 𝐵 → (𝐴 = ∅ ∧ 𝐵 = ∅))) |
11 | eqtr3 2752 | . 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵) | |
12 | 10, 11 | impbid1 224 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = 𝐵 ↔ (𝐴 = ∅ ∧ 𝐵 = ∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∩ cin 3946 ∅c0 4325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-in 3954 |
This theorem is referenced by: epnsym 9652 |
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