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Theorem compleq 4146
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 4145 . . 3 (𝐴𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴))
2 complss 4145 . . 3 (𝐵𝐴 ↔ (V ∖ 𝐴) ⊆ (V ∖ 𝐵))
31, 2anbi12ci 629 . 2 ((𝐴𝐵𝐵𝐴) ↔ ((V ∖ 𝐴) ⊆ (V ∖ 𝐵) ∧ (V ∖ 𝐵) ⊆ (V ∖ 𝐴)))
4 eqss 3996 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3996 . 2 ((V ∖ 𝐴) = (V ∖ 𝐵) ↔ ((V ∖ 𝐴) ⊆ (V ∖ 𝐵) ∧ (V ∖ 𝐵) ⊆ (V ∖ 𝐴)))
63, 4, 53bitr4i 303 1 (𝐴 = 𝐵 ↔ (V ∖ 𝐴) = (V ∖ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  Vcvv 3475  cdif 3944  wss 3947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3950  df-in 3954  df-ss 3964
This theorem is referenced by: (None)
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