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Theorem vdif0 4379
 Description: Universal class equality in terms of empty difference. (Contributed by NM, 17-Sep-2003.)
Assertion
Ref Expression
vdif0 (𝐴 = V ↔ (V ∖ 𝐴) = ∅)

Proof of Theorem vdif0
StepHypRef Expression
1 vss 4354 . 2 (V ⊆ 𝐴𝐴 = V)
2 ssdif0 4280 . 2 (V ⊆ 𝐴 ↔ (V ∖ 𝐴) = ∅)
31, 2bitr3i 280 1 (𝐴 = V ↔ (V ∖ 𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538  Vcvv 3444   ∖ cdif 3881   ⊆ wss 3884  ∅c0 4246 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 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-dif 3887  df-in 3891  df-ss 3901  df-nul 4247 This theorem is referenced by:  setind  9164
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