Theorem vdif0 4404

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 4381	. 2 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)
2		ssdif0 4301	. 2 ⊢ (V ⊆ 𝐴 ↔ (V ∖ 𝐴) = ∅)
3	1, 2	bitr3i 278	1 ⊢ (𝐴 = V ↔ (V ∖ 𝐴) = ∅)

Syntax hints:   ↔ wb 207   = wceq 1547  Vcvv 3432   ∖ cdif 3887   ⊆ wss 3890  ∅c0 4268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-dif 3893  df-ss 3907  df-nul 4269

This theorem is referenced by:  setind  9666  setindregs  35318