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Theorem dfop 4594
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 4593 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
41, 2, 3mp2an 675 1 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1637  wcel 2156  Vcvv 3391  {csn 4370  {cpr 4372  cop 4376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-v 3393  df-dif 3772  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-op 4377
This theorem is referenced by:  opi1  5126  opi2  5127  op1stb  5129  opeqsnOLD  5158  opeqpr  5159  propssopi  5163  uniop  5170  xpsspw  5434  relop  5474  funopg  6131
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