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Theorem opeluu 4447
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 3697 . . 3 𝐴 ∈ {𝐴, 𝐵}
3 opeluu.2 . . . . 5 𝐵 ∈ V
41, 3opi2 4230 . . . 4 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
5 elunii 3812 . . . 4 (({𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩ ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝐶)
64, 5mpan 424 . . 3 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → {𝐴, 𝐵} ∈ 𝐶)
7 elunii 3812 . . 3 ((𝐴 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ 𝐶) → 𝐴 𝐶)
82, 6, 7sylancr 414 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 𝐶)
93prid2 3698 . . 3 𝐵 ∈ {𝐴, 𝐵}
10 elunii 3812 . . 3 ((𝐵 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ 𝐶) → 𝐵 𝐶)
119, 6, 10sylancr 414 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐵 𝐶)
128, 11jca 306 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴 𝐶𝐵 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2148  Vcvv 2737  {cpr 3592  cop 3594   cuni 3807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808
This theorem is referenced by:  asymref  5010
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