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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 4180 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐵)) | |
2 | inidm 4194 | . . . . . 6 ⊢ (𝐵 ∩ 𝐵) = 𝐵 | |
3 | 1, 2 | syl6eq 2872 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐵) |
4 | 3 | eqeq1d 2823 | . . . 4 ⊢ (𝐴 = 𝐵 → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐵 = ∅)) |
5 | eqtr 2841 | . . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = ∅) → 𝐴 = ∅) | |
6 | simpr 485 | . . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = ∅) → 𝐵 = ∅) | |
7 | 5, 6 | jca 512 | . . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = ∅) → (𝐴 = ∅ ∧ 𝐵 = ∅)) |
8 | 7 | ex 413 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐵 = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅))) |
9 | 4, 8 | sylbid 241 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅))) |
10 | 9 | com12 32 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = 𝐵 → (𝐴 = ∅ ∧ 𝐵 = ∅))) |
11 | eqtr3 2843 | . 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵) | |
12 | 10, 11 | impbid1 226 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = 𝐵 ↔ (𝐴 = ∅ ∧ 𝐵 = ∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∩ cin 3934 ∅c0 4290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-in 3942 |
This theorem is referenced by: epnsym 9061 |
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