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Theorem symdifexg 4103
 Description: The symmetric difference of two sets is a set. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
symdifexg ((A V B W) → (AB) V)

Proof of Theorem symdifexg
StepHypRef Expression
1 df-symdif 3216 . 2 (AB) = ((A B) ∪ (B A))
2 difexg 4102 . . 3 ((A V B W) → (A B) V)
3 difexg 4102 . . . 4 ((B W A V) → (B A) V)
43ancoms 439 . . 3 ((A V B W) → (B A) V)
5 unexg 4101 . . 3 (((A B) V (B A) V) → ((A B) ∪ (B A)) V)
62, 4, 5syl2anc 642 . 2 ((A V B W) → ((A B) ∪ (B A)) V)
71, 6syl5eqel 2437 1 ((A V B W) → (AB) V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  Vcvv 2859   ∖ cdif 3206   ∪ cun 3207   ⊕ csymdif 3209 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-un 3214  df-dif 3215  df-symdif 3216 This theorem is referenced by:  symdifex  4108  imagekexg  4311  imageexg  5800  qsexg  5982
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