Theorem disj2 4412

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 3960	. 2 ⊢ 𝐴 ⊆ V
2		reldisj 4407	. 2 ⊢ (𝐴 ⊆ V → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))

Syntax hints:   ↔ wb 206   = wceq 1542  Vcvv 3442   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3444  df-dif 3906  df-in 3910  df-ss 3920  df-nul 4288

This theorem is referenced by:  ssindif0  4418  intirr  6083  setsres  17117  setscom  17119  f1omvdco3  19390  psgnunilem5  19435  opsrtoslem2  22023  clsconn  23386  cldsubg  24067  uniinn0  32636  imadifxp  32687