ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uneqri GIF version

Theorem uneqri 3275
Description: Inference from membership to union. (Contributed by NM, 5-Aug-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 3274 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2 uneqri.1 . . 3 ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)
31, 2bitri 184 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐶)
43eqriv 2172 1 (𝐴𝐵) = 𝐶
Colors of variables: wff set class
Syntax hints:  wb 105  wo 708   = wceq 1353  wcel 2146  cun 3125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131
This theorem is referenced by:  unidm  3276  uncom  3277  unass  3290  undi  3381  unab  3400  un0  3454
  Copyright terms: Public domain W3C validator