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Theorem vdif0 4014
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 3989 . 2 (V ⊆ 𝐴𝐴 = V)
2 ssdif0 3921 . 2 (V ⊆ 𝐴 ↔ (V ∖ 𝐴) = ∅)
31, 2bitr3i 266 1 (𝐴 = V ↔ (V ∖ 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  Vcvv 3191  cdif 3557  wss 3560  c0 3896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-dif 3563  df-in 3567  df-ss 3574  df-nul 3897
This theorem is referenced by:  setind  8555
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