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Theorem uneqri 3603
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 3601 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2 uneqri.1 . . 3 ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)
31, 2bitri 259 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐶)
43eqriv 2502 1 (𝐴𝐵) = 𝐶
Colors of variables: wff setvar class
Syntax hints:  wb 191  wo 377   = wceq 1468  wcel 1937  cun 3424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-v 3068  df-un 3431
This theorem is referenced by:  unidm  3604  uncom  3605  unass  3618  dfun2  3705  undi  3717  unab  3736  un0  3797  inundif  3872
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