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Theorem uneqri 3406
 Description: Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
uneqri.1 ((x A x B) ↔ x C)
Assertion
Ref Expression
uneqri (AB) = C
Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem uneqri
StepHypRef Expression
1 elun 3220 . . 3 (x (AB) ↔ (x A x B))
2 uneqri.1 . . 3 ((x A x B) ↔ x C)
31, 2bitri 240 . 2 (x (AB) ↔ x C)
43eqriv 2350 1 (AB) = C
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∨ wo 357   = wceq 1642   ∈ wcel 1710   ∪ 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 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:  unidm  3407  uncom  3408  unass  3420  dfun2  3490  undi  3502  unab  3521  un0  3575  inundif  3628
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