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

Theorem opeluu 4182
 Description: Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
opeluu.1
opeluu.2
Assertion
Ref Expression
opeluu

Proof of Theorem opeluu
StepHypRef Expression
1 opeluu.1 . . . 4
21prid1 3476 . . 3
3 opeluu.2 . . . . 5
41, 3opi2 3970 . . . 4
5 elunii 3585 . . . 4
64, 5mpan 400 . . 3
7 elunii 3585 . . 3
82, 6, 7sylancr 393 . 2
93prid2 3477 . . 3
10 elunii 3585 . . 3
119, 6, 10sylancr 393 . 2
128, 11jca 290 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wcel 1393  cvv 2557  cpr 3376  cop 3378  cuni 3580 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581 This theorem is referenced by:  asymref  4710
 Copyright terms: Public domain W3C validator