Theorem disj2 4433

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 3983	. 2 ⊢ 𝐴 ⊆ V
2		reldisj 4428	. 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 1540  Vcvv 3459   ∖ cdif 3923   ∩ cin 3925   ⊆ wss 3926  ∅c0 4308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-v 3461  df-dif 3929  df-in 3933  df-ss 3943  df-nul 4309

This theorem is referenced by:  ssindif0  4439  intirr  6107  setsres  17197  setscom  17199  f1omvdco3  19430  psgnunilem5  19475  opsrtoslem2  22014  clsconn  23368  cldsubg  24049  uniinn0  32531  imadifxp  32582