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Theorem dfopg 3538
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 2560 . 2 (A 𝑉A V)
2 elex 2560 . 2 (B 𝑊B V)
3 df-3an 886 . . . . . 6 ((A V B V x {{A}, {A, B}}) ↔ ((A V B V) x {{A}, {A, B}}))
43baibr 828 . . . . 5 ((A V B V) → (x {{A}, {A, B}} ↔ (A V B V x {{A}, {A, B}})))
54abbidv 2152 . . . 4 ((A V B V) → {xx {{A}, {A, B}}} = {x ∣ (A V B V x {{A}, {A, B}})})
6 abid2 2155 . . . 4 {xx {{A}, {A, B}}} = {{A}, {A, B}}
7 df-op 3376 . . . . 5 A, B⟩ = {x ∣ (A V B V x {{A}, {A, B}})}
87eqcomi 2041 . . . 4 {x ∣ (A V B V x {{A}, {A, B}})} = ⟨A, B
95, 6, 83eqtr3g 2092 . . 3 ((A V B V) → {{A}, {A, B}} = ⟨A, B⟩)
109eqcomd 2042 . 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 884   = wceq 1242   wcel 1390  {cab 2023  Vcvv 2551  {csn 3367  {cpr 3368  cop 3370
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553  df-op 3376
This theorem is referenced by:  dfop  3539  opexg  3955  opexgOLD  3956  opth1  3964  opth  3965  0nelop  3976  op1stbg  4176
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