Theorem difex 4107

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 4102	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A ∖ B) ∈ V)
4	1, 2, 3	mp2an 653	1 ⊢ (A ∖ B) ∈ V

Syntax hints:   ∈ wcel 1710  Vcvv 2859   ∖ cdif 3206

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  df-compl 3212  df-in 3213  df-dif 3215

This theorem is referenced by:  pwadjoin  4119  addcexlem  4382  nncex  4396  nnsucelrlem1  4424  nnsucelr  4428  ltfinex  4464  ssfin  4470  ncfinraiselem2  4480  ncfinlowerlem1  4482  tfinrelkex  4487  evenfinex  4503  oddfinex  4504  evenodddisjlem1  4515  nnadjoinlem1  4519  nnpweqlem1  4522  srelkex  4525  sfintfinlem1  4531  tfinnnlem1  4533  sfinltfin  4535  spfinex  4537  vfinspnn  4541  phialllem2  4617  phiall  4618  mpt2exlem  5811  funsex  5828  transex  5910  antisymex  5912  connexex  5913  foundex  5914  extex  5915  symex  5916  enadj  6060  ltcex  6116  2p1e3c  6156  sbthlem1  6203  dflec2  6210  nchoicelem11  6299  nchoicelem16  6304