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Theorem vdif0im 3398
Description: Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)
Assertion
Ref Expression
vdif0im (𝐴 = V → (V ∖ 𝐴) = ∅)

Proof of Theorem vdif0im
StepHypRef Expression
1 vss 3380 . 2 (V ⊆ 𝐴𝐴 = V)
2 ssdif0im 3397 . 2 (V ⊆ 𝐴 → (V ∖ 𝐴) = ∅)
31, 2sylbir 134 1 (𝐴 = V → (V ∖ 𝐴) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  Vcvv 2660  cdif 3038  wss 3041  c0 3333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054  df-nul 3334
This theorem is referenced by: (None)
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