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Theorem dfop 3699
Description: Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998.)
Hypotheses
Ref Expression
dfop.1 𝐴 ∈ V
dfop.2 𝐵 ∈ V
Assertion
Ref Expression
dfop 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}

Proof of Theorem dfop
StepHypRef Expression
1 dfop.1 . 2 𝐴 ∈ V
2 dfop.2 . 2 𝐵 ∈ V
3 dfopg 3698 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
41, 2, 3mp2an 422 1 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
Colors of variables: wff set class
Syntax hints:   = wceq 1331  wcel 1480  Vcvv 2681  {csn 3522  {cpr 3523  cop 3525
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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-v 2683  df-op 3531
This theorem is referenced by:  opid  3718  elop  4148  opi1  4149  opi2  4150  opeqsn  4169  opeqpr  4170  uniop  4172  op1stb  4394  xpsspw  4646  relop  4684  funopg  5152
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