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Theorem vdif0 4070
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 4045 . 2 (V ⊆ 𝐴𝐴 = V)
2 ssdif0 3975 . 2 (V ⊆ 𝐴 ↔ (V ∖ 𝐴) = ∅)
31, 2bitr3i 266 1 (𝐴 = V ↔ (V ∖ 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  Vcvv 3231  cdif 3604  wss 3607  c0 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949
This theorem is referenced by:  setind  8648
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