 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  vdif0 Structured version   Visualization version   GIF version

Theorem vdif0 3989
 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 3964 . 2 (V ⊆ 𝐴𝐴 = V)
2 ssdif0 3896 . 2 (V ⊆ 𝐴 ↔ (V ∖ 𝐴) = ∅)
31, 2bitr3i 265 1 (𝐴 = V ↔ (V ∖ 𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875 This theorem is referenced by:  setind  8493
 Copyright terms: Public domain W3C validator