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Theorem opeluu 5129
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 4486 . . 3 𝐴 ∈ {𝐴, 𝐵}
3 opeluu.2 . . . . 5 𝐵 ∈ V
41, 3opi2 5128 . . . 4 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
5 elunii 4633 . . . 4 (({𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩ ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝐶)
64, 5mpan 682 . . 3 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → {𝐴, 𝐵} ∈ 𝐶)
7 elunii 4633 . . 3 ((𝐴 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ 𝐶) → 𝐴 𝐶)
82, 6, 7sylancr 582 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 𝐶)
93prid2 4487 . . 3 𝐵 ∈ {𝐴, 𝐵}
10 elunii 4633 . . 3 ((𝐵 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ 𝐶) → 𝐵 𝐶)
119, 6, 10sylancr 582 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐵 𝐶)
128, 11jca 508 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴 𝐶𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wcel 2157  Vcvv 3385  {cpr 4370  cop 4374   cuni 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629
This theorem is referenced by:  asymref  5730  asymref2  5731  wrdexb  13545
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