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Theorem opeluu 5470
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 𝐶𝐵 𝐶))

Proof of Theorem opeluu
StepHypRef Expression
1 opeluu.1 . . . 4 𝐴 ∈ V
21prid1 4766 . . 3 𝐴 ∈ {𝐴, 𝐵}
3 opeluu.2 . . . . 5 𝐵 ∈ V
41, 3opi2 5469 . . . 4 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
5 elunii 4913 . . . 4 (({𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩ ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝐶)
64, 5mpan 688 . . 3 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → {𝐴, 𝐵} ∈ 𝐶)
7 elunii 4913 . . 3 ((𝐴 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ 𝐶) → 𝐴 𝐶)
82, 6, 7sylancr 587 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 𝐶)
93prid2 4767 . . 3 𝐵 ∈ {𝐴, 𝐵}
10 elunii 4913 . . 3 ((𝐵 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ 𝐶) → 𝐵 𝐶)
119, 6, 10sylancr 587 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐵 𝐶)
128, 11jca 512 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴 𝐶𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  Vcvv 3474  {cpr 4630  cop 4634   cuni 4908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909
This theorem is referenced by:  asymref  6117  asymref2  6118  wrdexb  14479
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