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Theorem uneqri 4155
Description: Inference from membership to union. (Contributed by NM, 21-Jun-1993.)
Hypothesis
Ref Expression
uneqri.1 ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)
Assertion
Ref Expression
uneqri (𝐴𝐵) = 𝐶
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem uneqri
StepHypRef Expression
1 elun 4152 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2 uneqri.1 . . 3 ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)
31, 2bitri 275 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐶)
43eqriv 2733 1 (𝐴𝐵) = 𝐶
Colors of variables: wff setvar class
Syntax hints:  wb 206  wo 847   = wceq 1539  wcel 2107  cun 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3955
This theorem is referenced by:  unidm  4156  uncom  4157  unass  4171  dfun2  4269  undi  4284  unab  4307  un0  4393  inundif  4478
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