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| 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 4150 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴)) | |
| 2 | complss 4150 | . . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (V ∖ 𝐴) ⊆ (V ∖ 𝐵)) | |
| 3 | 1, 2 | anbi12ci 629 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ ((V ∖ 𝐴) ⊆ (V ∖ 𝐵) ∧ (V ∖ 𝐵) ⊆ (V ∖ 𝐴))) | 
| 4 | eqss 3998 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
| 5 | eqss 3998 | . 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 206 ∧ wa 395 = wceq 1539 Vcvv 3479 ∖ cdif 3947 ⊆ wss 3950 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-dif 3953 df-ss 3967 | 
| This theorem is referenced by: (None) | 
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