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Mirrors > Home > MPE Home > Th. List > compleq | Structured version Visualization version GIF version |
Description: Two classes are equal if and only if their complements are equal. (Contributed by BJ, 19-Mar-2021.) |
Ref | Expression |
---|---|
compleq | ⊢ (𝐴 = 𝐵 ↔ (V ∖ 𝐴) = (V ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | complss 4146 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴)) | |
2 | complss 4146 | . . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (V ∖ 𝐴) ⊆ (V ∖ 𝐵)) | |
3 | 1, 2 | anbi12ci 627 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ ((V ∖ 𝐴) ⊆ (V ∖ 𝐵) ∧ (V ∖ 𝐵) ⊆ (V ∖ 𝐴))) |
4 | eqss 3997 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | eqss 3997 | . 2 ⊢ ((V ∖ 𝐴) = (V ∖ 𝐵) ↔ ((V ∖ 𝐴) ⊆ (V ∖ 𝐵) ∧ (V ∖ 𝐵) ⊆ (V ∖ 𝐴))) | |
6 | 3, 4, 5 | 3bitr4i 303 | 1 ⊢ (𝐴 = 𝐵 ↔ (V ∖ 𝐴) = (V ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1540 Vcvv 3473 ∖ cdif 3945 ⊆ wss 3948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-dif 3951 df-in 3955 df-ss 3965 |
This theorem is referenced by: (None) |
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