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Theorem vdif0 4464
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 4439 . 2 (V ⊆ 𝐴𝐴 = V)
2 ssdif0 4359 . 2 (V ⊆ 𝐴 ↔ (V ∖ 𝐴) = ∅)
31, 2bitr3i 277 1 (𝐴 = V ↔ (V ∖ 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  Vcvv 3470  cdif 3942  wss 3945  c0 4318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-dif 3948  df-in 3952  df-ss 3962  df-nul 4319
This theorem is referenced by:  setind  9751
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