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Theorem opeluu 4210
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 3504 . . 3 𝐴 ∈ {𝐴, 𝐵}
3 opeluu.2 . . . . 5 𝐵 ∈ V
41, 3opi2 3998 . . . 4 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
5 elunii 3613 . . . 4 (({𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩ ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝐶)
64, 5mpan 408 . . 3 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → {𝐴, 𝐵} ∈ 𝐶)
7 elunii 3613 . . 3 ((𝐴 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ 𝐶) → 𝐴 𝐶)
82, 6, 7sylancr 399 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 𝐶)
93prid2 3505 . . 3 𝐵 ∈ {𝐴, 𝐵}
10 elunii 3613 . . 3 ((𝐵 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ 𝐶) → 𝐵 𝐶)
119, 6, 10sylancr 399 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐵 𝐶)
128, 11jca 294 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴 𝐶𝐵 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  Vcvv 2574  {cpr 3404  cop 3406   cuni 3608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609
This theorem is referenced by:  asymref  4738
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