Theorem difex 4108

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

Hypotheses

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

Assertion

Ref	Expression
difex	⊢ (A ∖ B) ∈ V

Proof of Theorem difex

Step	Hyp	Ref	Expression
1		boolex.1	. 2 ⊢ A ∈ V
2		boolex.2	. 2 ⊢ B ∈ V
3		difexg 4103	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A ∖ B) ∈ V)
4	1, 2, 3	mp2an 653	1 ⊢ (A ∖ B) ∈ V

Syntax hints:   ∈ wcel 1710  Vcvv 2860   ∖ cdif 3207

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 4079

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216

This theorem is referenced by:  pwadjoin  4120  addcexlem  4383  nncex  4397  nnsucelrlem1  4425  nnsucelr  4429  ltfinex  4465  ssfin  4471  ncfinraiselem2  4481  ncfinlowerlem1  4483  tfinrelkex  4488  evenfinex  4504  oddfinex  4505  evenodddisjlem1  4516  nnadjoinlem1  4520  nnpweqlem1  4523  srelkex  4526  sfintfinlem1  4532  tfinnnlem1  4534  sfinltfin  4536  spfinex  4538  vfinspnn  4542  phialllem2  4618  phiall  4619  mpt2exlem  5812  funsex  5829  transex  5911  antisymex  5913  connexex  5914  foundex  5915  extex  5916  symex  5917  enadj  6061  ltcex  6117  2p1e3c  6157  sbthlem1  6204  dflec2  6211  nchoicelem11  6300  nchoicelem16  6305