Theorem disj2 4347

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 3901	. 2 ⊢ 𝐴 ⊆ V
2		reldisj 4341	. 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 1542  Vcvv 3398   ∖ cdif 3840   ∩ cin 3842   ⊆ wss 3843  ∅c0 4211

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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-v 3400  df-dif 3846  df-in 3850  df-ss 3860  df-nul 4212

This theorem is referenced by:  ssindif0  4353  intirr  5952  setsres  16628  setscom  16630  f1omvdco3  18695  psgnunilem5  18740  opsrtoslem2  20867  clsconn  22181  cldsubg  22862  uniinn0  30464  imadifxp  30514