Theorem disj2 4419

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 3962	. 2 ⊢ 𝐴 ⊆ V
2		reldisj 4414	. 2 ⊢ (𝐴 ⊆ V → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))

Syntax hints:   ↔ wb 209   = wceq 1570  Vcvv 3455   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-v 3457  df-dif 3909  df-in 3913  df-ss 3923  df-nul 4288

This theorem is referenced by:  ssindif0  4425  intirr  6120  setsres  17239  setscom  17241  f1omvdco3  19520  psgnunilem5  19565  opsrtoslem2  22188  clsconn  23568  cldsubg  24249  uniinn0  32875  imadifxp  32924