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Theorem opeluu 4417
 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 3638 . . 3
3 opeluu.2 . . . . 5
41, 3opi2 4134 . . . 4
5 elunii 3732 . . . 4
64, 5mpan 654 . . 3
7 elunii 3732 . . 3
82, 6, 7sylancr 647 . 2
93prid2 3639 . . 3
10 elunii 3732 . . 3
119, 6, 10sylancr 647 . 2
128, 11jca 520 1
 Colors of variables: wff set class Syntax hints:   wi 6   wa 360   wcel 1621  cvv 2727  cpr 3545  cop 3547  cuni 3727 This theorem is referenced by:  asymref  4966  asymref2  4967  wrdexb  11326  dfdir2  24457 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728
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