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Theorem dfop 4762
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.) (Avoid depending on this detail.)
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 4761 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
41, 2, 3mp2an 691 1 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2111  Vcvv 3441  {csn 4525  {cpr 4527  cop 4531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-op 4532
This theorem is referenced by:  opi1  5325  opi2  5326  op1stb  5328  opeqpr  5360  propssopi  5363  uniop  5370  xpsspw  5646  relop  5685  funopg  6358
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