Theorem vdif0im 3394

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 3376	. 2 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)
2		ssdif0im 3393	. 2 ⊢ (V ⊆ 𝐴 → (V ∖ 𝐴) = ∅)
3	1, 2	sylbir 134	1 ⊢ (𝐴 = V → (V ∖ 𝐴) = ∅)

Syntax hints:   → wi 4   = wceq 1314  Vcvv 2657   ∖ cdif 3034   ⊆ wss 3037  ∅c0 3329

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-dif 3039  df-in 3043  df-ss 3050  df-nul 3330

This theorem is referenced by: (None)