Theorem disj2 4422

Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)

Assertion

Ref	Expression
disj2	⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))

Proof of Theorem disj2

Step	Hyp	Ref	Expression
1		ssv 3971	. 2 ⊢ 𝐴 ⊆ V
2		reldisj 4416	. 2 ⊢ (𝐴 ⊆ V → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))

Syntax hints:   ↔ wb 205   = wceq 1541  Vcvv 3446   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-v 3448  df-dif 3916  df-in 3920  df-ss 3930  df-nul 4288

This theorem is referenced by:  ssindif0  4428  intirr  6077  setsres  17061  setscom  17063  f1omvdco3  19245  psgnunilem5  19290  opsrtoslem2  21500  clsconn  22818  cldsubg  23499  uniinn0  31536  imadifxp  31586