Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  compleq Structured version   Visualization version   GIF version

Theorem compleq 4053
 Description: Two classes are equal if and only if their complements are equal. (Contributed by BJ, 19-Mar-2021.)
Assertion
Ref Expression
compleq (𝐴 = 𝐵 ↔ (V ∖ 𝐴) = (V ∖ 𝐵))

Proof of Theorem compleq
StepHypRef Expression
1 complss 4052 . . 3 (𝐴𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴))
2 complss 4052 . . 3 (𝐵𝐴 ↔ (V ∖ 𝐴) ⊆ (V ∖ 𝐵))
31, 2anbi12ci 630 . 2 ((𝐴𝐵𝐵𝐴) ↔ ((V ∖ 𝐴) ⊆ (V ∖ 𝐵) ∧ (V ∖ 𝐵) ⊆ (V ∖ 𝐴)))
4 eqss 3907 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3907 . 2 ((V ∖ 𝐴) = (V ∖ 𝐵) ↔ ((V ∖ 𝐴) ⊆ (V ∖ 𝐵) ∧ (V ∖ 𝐵) ⊆ (V ∖ 𝐴)))
63, 4, 53bitr4i 306 1 (𝐴 = 𝐵 ↔ (V ∖ 𝐴) = (V ∖ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  Vcvv 3409   ∖ cdif 3855   ⊆ wss 3858 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-dif 3861  df-in 3865  df-ss 3875 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator