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Theorem opelvv 4410
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 4396 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
41, 2, 3mp2an 417 1 𝐴, 𝐵⟩ ∈ (V × V)
Colors of variables: wff set class
Syntax hints:  wcel 1434  Vcvv 2602  cop 3403   × cxp 4363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-opab 3842  df-xp 4371
This theorem is referenced by:  relsnop  4466  relopabi  4485  eqop2  5829
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