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Theorem opelvv 5716
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 5713 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
41, 2, 3mp2an 689 1 𝐴, 𝐵⟩ ∈ (V × V)
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  Vcvv 3473  cop 4634   × cxp 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-opab 5211  df-xp 5682
This theorem is referenced by:  relopabiALT  5823  funsneqopb  7152  isof1oopb  7325  1st2ndb  8018  eqop2  8021  evlfcl  18180  brtxp  35157  brpprod  35162  brsset  35166  brcart  35209  brcup  35216  brcap  35217  elcnvlem  42655
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