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Theorem opelvv 4596
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 4578 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
41, 2, 3mp2an 423 1 𝐴, 𝐵⟩ ∈ (V × V)
Colors of variables: wff set class
Syntax hints:  wcel 1481  Vcvv 2689  cop 3534   × cxp 4544
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-opab 3997  df-xp 4552
This theorem is referenced by:  relsnop  4652  relopabi  4672  eqop2  6083
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