Theorem vdif0 4492

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 4469	. 2 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)
2		ssdif0 4389	. 2 ⊢ (V ⊆ 𝐴 ↔ (V ∖ 𝐴) = ∅)
3	1, 2	bitr3i 277	1 ⊢ (𝐴 = V ↔ (V ∖ 𝐴) = ∅)

Syntax hints:   ↔ wb 206   = wceq 1537  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-ss 3993  df-nul 4353

This theorem is referenced by:  setind  9803