Theorem vdif0 4430

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 4366	. 2 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)
2		ssdif0 4322	. 2 ⊢ (V ⊆ 𝐴 ↔ (V ∖ 𝐴) = ∅)
3	1, 2	bitr3i 280	1 ⊢ (𝐴 = V ↔ (V ∖ 𝐴) = ∅)

Syntax hints:   ↔ wb 209   = wceq 1570  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-dif 3909  df-ss 3923  df-nul 4288

This theorem is referenced by:  setind  9717  setindregs  35524