Theorem disj2 4393

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 3946	. 2 ⊢ 𝐴 ⊆ V
2		reldisj 4388	. 2 ⊢ (𝐴 ⊆ V → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))

Syntax hints:   ↔ wb 207   = wceq 1547  Vcvv 3432   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-v 3434  df-dif 3893  df-in 3897  df-ss 3907  df-nul 4269

This theorem is referenced by:  ssindif0  4399  intirr  6075  setsres  17146  setscom  17148  f1omvdco3  19422  psgnunilem5  19467  opsrtoslem2  22039  clsconn  23420  cldsubg  24101  uniinn0  32646  imadifxp  32697