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Theorem dfopg 4368
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 3198 . 2 (𝐴𝑉𝐴 ∈ V)
2 elex 3198 . 2 (𝐵𝑊𝐵 ∈ V)
3 dfopif 4367 . . 3 𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
4 iftrue 4064 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) = {{𝐴}, {𝐴, 𝐵}})
53, 4syl5eq 2667 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
61, 2, 5syl2an 494 1 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  c0 3891  ifcif 4058  {csn 4148  {cpr 4150  cop 4154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-op 4155
This theorem is referenced by:  dfop  4369  elopg  4895  opnz  4902  opth1  4904  opth  4905  0nelop  4920  opwf  8619  rankopb  8659  wunop  9488  tskop  9537  gruop  9571  opidg  40594
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