Theorem symdifexg 4104

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

Assertion

Ref	Expression
symdifexg	⊢ ((A ∈ V ∧ B ∈ W) → (A ⊕ B) ∈ V)

Proof of Theorem symdifexg

Step	Hyp	Ref	Expression
1		df-symdif 3217	. 2 ⊢ (A ⊕ B) = ((A ∖ B) ∪ (B ∖ A))
2		difexg 4103	. . 3 ⊢ ((A ∈ V ∧ B ∈ W) → (A ∖ B) ∈ V)
3		difexg 4103	. . . 4 ⊢ ((B ∈ W ∧ A ∈ V) → (B ∖ A) ∈ V)
4	3	ancoms 439	. . 3 ⊢ ((A ∈ V ∧ B ∈ W) → (B ∖ A) ∈ V)
5		unexg 4102	. . 3 ⊢ (((A ∖ B) ∈ V ∧ (B ∖ A) ∈ V) → ((A ∖ B) ∪ (B ∖ A)) ∈ V)
6	2, 4, 5	syl2anc 642	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → ((A ∖ B) ∪ (B ∖ A)) ∈ V)
7	1, 6	syl5eqel 2437	1 ⊢ ((A ∈ V ∧ B ∈ W) → (A ⊕ B) ∈ V)

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  Vcvv 2860   ∖ cdif 3207   ∪ cun 3208   ⊕ 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:  symdifex  4109  imagekexg  4312  imageexg  5801  qsexg  5983