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Theorem dfopg 4671
 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 3426 . 2 (𝐴𝑉𝐴 ∈ V)
2 elex 3426 . 2 (𝐵𝑊𝐵 ∈ V)
3 dfopif 4670 . . 3 𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
4 iftrue 4350 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) = {{𝐴}, {𝐴, 𝐵}})
53, 4syl5eq 2819 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
61, 2, 5syl2an 587 1 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∈ wcel 2051  Vcvv 3408  ∅c0 4172  ifcif 4344  {csn 4435  {cpr 4437  ⟨cop 4441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-v 3410  df-dif 3825  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-op 4442 This theorem is referenced by:  dfop  4672  opidg  4692  elopg  5211  opnz  5218  opth1  5220  opth  5221  0nelop  5238  opeqsng  5245  opwf  9033  rankopb  9073  wunop  9940  tskop  9989  gruop  10023
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