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Theorem uniopel 5383
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 5382 . . 3 𝐴, 𝐵⟩ = {𝐴, 𝐵}
41, 2opi2 5338 . . 3 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
53, 4eqeltri 2910 . 2 𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵
6 elssuni 4843 . . 3 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ⊆ 𝐶)
76sseld 3941 . 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 2114  Vcvv 3469  {cpr 4541  cop 4545   cuni 4813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814
This theorem is referenced by:  dmrnssfld  5819  unielrel  6103
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