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Theorem unexg 4101
 Description: The union of two sets is a set. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
unexg ((A V B W) → (AB) V)

Proof of Theorem unexg
StepHypRef Expression
1 df-un 3214 . 2 (AB) = ( ∼ A ⩃ ∼ B)
2 complexg 4099 . . 3 (A V → ∼ A V)
3 complexg 4099 . . 3 (B W → ∼ B V)
4 ninexg 4097 . . 3 (( ∼ A V B V) → ( ∼ A ⩃ ∼ B) V)
52, 3, 4syl2an 463 . 2 ((A V B W) → ( ∼ A ⩃ ∼ B) V)
61, 5syl5eqel 2437 1 ((A V B W) → (AB) V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  Vcvv 2859   ⩃ cnin 3204   ∼ ccompl 3205   ∪ cun 3207 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-un 3214 This theorem is referenced by:  symdifexg  4103  unex  4106  imakexg  4299  ncfindi  4475  opexg  4587  cupvalg  5812  fullfunexg  5859  addccan2nclem2  6264
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