Theorem vdif0im 3512

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 3494	. 2 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)
2		ssdif0im 3511	. 2 ⊢ (V ⊆ 𝐴 → (V ∖ 𝐴) = ∅)
3	1, 2	sylbir 135	1 ⊢ (𝐴 = V → (V ∖ 𝐴) = ∅)

Syntax hints:   → wi 4   = wceq 1364  Vcvv 2760   ∖ cdif 3150   ⊆ wss 3153  ∅c0 3446

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447

This theorem is referenced by: (None)