Theorem symdifex 4109

Description: The symmetric 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
symdifex	⊢ (A ⊕ B) ∈ V

Proof of Theorem symdifex

Step	Hyp	Ref	Expression
1		boolex.1	. 2 ⊢ A ∈ V
2		boolex.2	. 2 ⊢ B ∈ V
3		symdifexg 4104	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A ⊕ B) ∈ V)
4	1, 2, 3	mp2an 653	1 ⊢ (A ⊕ B) ∈ V

Syntax hints:   ∈ wcel 1710  Vcvv 2860   ⊕ csymdif 3210

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-un 3215  df-dif 3216  df-symdif 3217

This theorem is referenced by:  addcexlem  4383  nnsucelrlem1  4425  ltfinex  4465  ncfinraiselem2  4481  ncfinlowerlem1  4483  tfinrelkex  4488  evenfinex  4504  oddfinex  4505  evenodddisjlem1  4516  nnadjoinlem1  4520  nnpweqlem1  4523  srelkex  4526  tfinnnlem1  4534  opexg  4588  proj2exg  4593  setconslem5  4736  1stex  4740  swapex  4743  mptexlem  5811  mpt2exlem  5812  extex  5916  ovcelem1  6172  ceex  6175  tcfnex  6245  nmembers1lem1  6269  nchoicelem11  6300  nchoicelem16  6305  nchoicelem18  6307