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Theorem uniopel 4148
Description: Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opthw.1 𝐴 ∈ V
opthw.2 𝐵 ∈ V
Assertion
Ref Expression
uniopel (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴, 𝐵⟩ ∈ 𝐶)

Proof of Theorem uniopel
StepHypRef Expression
1 opthw.1 . . . 4 𝐴 ∈ V
2 opthw.2 . . . 4 𝐵 ∈ V
31, 2uniop 4147 . . 3 𝐴, 𝐵⟩ = {𝐴, 𝐵}
41, 2opi2 4125 . . 3 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
53, 4eqeltri 2190 . 2 𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵
6 elssuni 3734 . . 3 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ⊆ 𝐶)
76sseld 3066 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵⟩ → 𝐴, 𝐵⟩ ∈ 𝐶))
85, 7mpi 15 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴, 𝐵⟩ ∈ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1465  Vcvv 2660  {cpr 3498  cop 3500   cuni 3706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707
This theorem is referenced by:  dmrnssfld  4772  unielrel  5036
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