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Theorem uniopel 4216
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 4215 . . 3 𝐴, 𝐵⟩ = {𝐴, 𝐵}
41, 2opi2 4193 . . 3 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
53, 4eqeltri 2230 . 2 𝐴, 𝐵⟩ ∈ ⟨𝐴, 𝐵
6 elssuni 3800 . . 3 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ⊆ 𝐶)
76sseld 3127 . 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 2128  Vcvv 2712  {cpr 3561  cop 3563   cuni 3772
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773
This theorem is referenced by:  dmrnssfld  4848  unielrel  5112
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