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Theorem opelvv 5562
Description: Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opelvv.1 𝐴 ∈ V
opelvv.2 𝐵 ∈ V
Assertion
Ref Expression
opelvv 𝐴, 𝐵⟩ ∈ (V × V)

Proof of Theorem opelvv
StepHypRef Expression
1 opelvv.1 . 2 𝐴 ∈ V
2 opelvv.2 . 2 𝐵 ∈ V
3 opelxpi 5560 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
41, 2, 3mp2an 691 1 𝐴, 𝐵⟩ ∈ (V × V)
Colors of variables: wff setvar class
Syntax hints:  wcel 2112  Vcvv 3444  cop 4534   × cxp 5521
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
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 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-opab 5096  df-xp 5529
This theorem is referenced by:  relopabiALT  5663  funsneqopb  6895  isof1oopb  7061  1st2ndb  7715  eqop2  7718  evlfcl  17468  brtxp  33455  brpprod  33460  brsset  33464  brcart  33507  brcup  33514  brcap  33515  elcnvlem  40294
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