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Theorem opelvvg 5087
Description: Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)
Assertion
Ref Expression
opelvvg ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))

Proof of Theorem opelvvg
StepHypRef Expression
1 elex 3185 . 2 (𝐴𝑉𝐴 ∈ V)
2 elex 3185 . 2 (𝐵𝑊𝐵 ∈ V)
3 opelxpi 5072 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
41, 2, 3syl2an 493 1 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 1977  Vcvv 3173  cop 4131   × cxp 5036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044
This theorem is referenced by:  isof1oopb  6475  opvtxfv  25681  opiedgfv  25684
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