Theorem vdif0 4461

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

Step	Hyp	Ref	Expression
1		vss 4436	. 2 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)
2		ssdif0 4356	. 2 ⊢ (V ⊆ 𝐴 ↔ (V ∖ 𝐴) = ∅)
3	1, 2	bitr3i 277	1 ⊢ (𝐴 = V ↔ (V ∖ 𝐴) = ∅)

Syntax hints:   ↔ wb 205   = wceq 1533  Vcvv 3466   ∖ cdif 3938   ⊆ wss 3941  ∅c0 4315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-dif 3944  df-in 3948  df-ss 3958  df-nul 4316

This theorem is referenced by:  setind  9726