Theorem vdif0 4295

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 4272	. 2 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)
2		ssdif0 4203	. 2 ⊢ (V ⊆ 𝐴 ↔ (V ∖ 𝐴) = ∅)
3	1, 2	bitr3i 269	1 ⊢ (𝐴 = V ↔ (V ∖ 𝐴) = ∅)

Syntax hints:   ↔ wb 198   = wceq 1508  Vcvv 3408   ∖ cdif 3819   ⊆ wss 3822  ∅c0 4172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-v 3410  df-dif 3825  df-in 3829  df-ss 3836  df-nul 4173

This theorem is referenced by:  setind  8968