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Theorem opeluu 4148
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 A V
opeluu.2 B V
Assertion
Ref Expression
opeluu (⟨A, B 𝐶 → (A 𝐶 B 𝐶))

Proof of Theorem opeluu
StepHypRef Expression
1 opeluu.1 . . . 4 A V
21prid1 3467 . . 3 A {A, B}
3 opeluu.2 . . . . 5 B V
41, 3opi2 3961 . . . 4 {A, B} A, B
5 elunii 3576 . . . 4 (({A, B} A, BA, B 𝐶) → {A, B} 𝐶)
64, 5mpan 400 . . 3 (⟨A, B 𝐶 → {A, B} 𝐶)
7 elunii 3576 . . 3 ((A {A, B} {A, B} 𝐶) → A 𝐶)
82, 6, 7sylancr 393 . 2 (⟨A, B 𝐶A 𝐶)
93prid2 3468 . . 3 B {A, B}
10 elunii 3576 . . 3 ((B {A, B} {A, B} 𝐶) → B 𝐶)
119, 6, 10sylancr 393 . 2 (⟨A, B 𝐶B 𝐶)
128, 11jca 290 1 (⟨A, B 𝐶 → (A 𝐶 B 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  Vcvv 2551  {cpr 3368  cop 3370   cuni 3571
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572
This theorem is referenced by:  asymref  4653
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