Theorem vdif0 3610

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 3587	. 2 ⊢ (V ⊆ A ↔ A = V)
2		ssdif0 3609	. 2 ⊢ (V ⊆ A ↔ (V ∖ A) = ∅)
3	1, 2	bitr3i 242	1 ⊢ (A = V ↔ (V ∖ A) = ∅)

Syntax hints:   ↔ wb 176   = wceq 1642  Vcvv 2859   ∖ cdif 3206   ⊆ wss 3257  ∅c0 3550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551

This theorem is referenced by: (None)