Theorem disj2 4411

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 4406	. 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 3438   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-v 3440  df-dif 3908  df-in 3912  df-ss 3922  df-nul 4287

This theorem is referenced by:  ssindif0  4417  intirr  6071  setsres  17107  setscom  17109  f1omvdco3  19346  psgnunilem5  19391  opsrtoslem2  21979  clsconn  23333  cldsubg  24014  uniinn0  32512  imadifxp  32563