Theorem vdif0im 3557

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

Step	Hyp	Ref	Expression
1		vss 3539	. 2 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)
2		ssdif0im 3556	. 2 ⊢ (V ⊆ 𝐴 → (V ∖ 𝐴) = ∅)
3	1, 2	sylbir 135	1 ⊢ (𝐴 = V → (V ∖ 𝐴) = ∅)

Syntax hints:   → wi 4   = wceq 1395  Vcvv 2799   ∖ cdif 3194   ⊆ wss 3197  ∅c0 3491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492

This theorem is referenced by: (None)