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Theorem uniopel 5397
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 5396 . . 3 𝐴, 𝐵⟩ = {𝐴, 𝐵}
41, 2opi2 5352 . . 3 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
53, 4eqeltri 2906 . 2 𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵
6 elssuni 4859 . . 3 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ⊆ 𝐶)
76sseld 3963 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵⟩ → 𝐴, 𝐵⟩ ∈ 𝐶))
85, 7mpi 20 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴, 𝐵⟩ ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  Vcvv 3492  {cpr 4559  cop 4563   cuni 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831
This theorem is referenced by:  dmrnssfld  5834  unielrel  6118
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