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Theorem dfopg 3517
Description: Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
dfopg ((A 𝑉 B 𝑊) → ⟨A, B⟩ = {{A}, {A, B}})

Proof of Theorem dfopg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elex 2539 . 2 (A 𝑉A V)
2 elex 2539 . 2 (B 𝑊B V)
3 df-3an 873 . . . . . 6 ((A V B V x {{A}, {A, B}}) ↔ ((A V B V) x {{A}, {A, B}}))
43baibr 817 . . . . 5 ((A V B V) → (x {{A}, {A, B}} ↔ (A V B V x {{A}, {A, B}})))
54abbidv 2133 . . . 4 ((A V B V) → {xx {{A}, {A, B}}} = {x ∣ (A V B V x {{A}, {A, B}})})
6 abid2 2136 . . . 4 {xx {{A}, {A, B}}} = {{A}, {A, B}}
7 df-op 3355 . . . . 5 A, B⟩ = {x ∣ (A V B V x {{A}, {A, B}})}
87eqcomi 2022 . . . 4 {x ∣ (A V B V x {{A}, {A, B}})} = ⟨A, B
95, 6, 83eqtr3g 2073 . . 3 ((A V B V) → {{A}, {A, B}} = ⟨A, B⟩)
109eqcomd 2023 . 2 ((A V B V) → ⟨A, B⟩ = {{A}, {A, B}})
111, 2, 10syl2an 273 1 ((A 𝑉 B 𝑊) → ⟨A, B⟩ = {{A}, {A, B}})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 871   = wceq 1226   wcel 1370  {cab 2004  Vcvv 2531  {csn 3346  {cpr 3347  cop 3349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-v 2533  df-op 3355
This theorem is referenced by:  dfop  3518  opexg  3934  opexgOLD  3935  opth1  3943  opth  3944  0nelop  3955  op1stbg  4156
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