Theorem ninex 4098

Description: The anti-intersection of two sets is a set. (Contributed by SF, 12-Jan-2015.)

Hypotheses

Ref	Expression
ninex.1	⊢ A ∈ V
ninex.2	⊢ B ∈ V

Assertion

Ref	Expression
ninex	⊢ (A ⩃ B) ∈ V

Proof of Theorem ninex

Step	Hyp	Ref	Expression
1		ninex.1	. 2 ⊢ A ∈ V
2		ninex.2	. 2 ⊢ B ∈ V
3		ninexg 4097	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A ⩃ B) ∈ V)
4	1, 2, 3	mp2an 653	1 ⊢ (A ⩃ B) ∈ V

Syntax hints:   ∈ wcel 1710  Vcvv 2859   ⩃ cnin 3204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211

This theorem is referenced by: (None)