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Theorem dfop 3616
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 3615 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
41, 2, 3mp2an 417 1 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wcel 1438  Vcvv 2619  {csn 3441  {cpr 3442  cop 3444
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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621  df-op 3450
This theorem is referenced by:  opid  3635  elop  4049  opi1  4050  opi2  4051  opeqsn  4070  opeqpr  4071  uniop  4073  op1stb  4290  xpsspw  4538  relop  4574  funopg  5034
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