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Theorem dfopg 4815
Description: Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
Assertion
Ref Expression
dfopg ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})

Proof of Theorem dfopg
StepHypRef Expression
1 elex 3459 . 2 (𝐴𝑉𝐴 ∈ V)
2 elex 3459 . 2 (𝐵𝑊𝐵 ∈ V)
3 dfopif 4814 . . 3 𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
4 iftrue 4479 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) = {{𝐴}, {𝐴, 𝐵}})
53, 4eqtrid 2788 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
61, 2, 5syl2an 596 1 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  Vcvv 3441  c0 4269  ifcif 4473  {csn 4573  {cpr 4575  cop 4579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-dif 3901  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-op 4580
This theorem is referenced by:  dfop  4816  opidg  4836  elopg  5411  opnz  5418  opth1  5420  opth  5421  0nelop  5440  opeqsng  5447  opwf  9669  rankopb  9709  wunop  10579  tskop  10628  gruop  10662
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