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Theorem opexgOLD 3956
Description: An ordered pair of sets is a set. This is a special case of opexg 3955 and new proofs should use opexg 3955 instead. (Contributed by Jim Kingdon, 19-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of opexg 3955 and then remove it.
Assertion
Ref Expression
opexgOLD ((A V B V) → ⟨A, B V)

Proof of Theorem opexgOLD
StepHypRef Expression
1 dfopg 3538 . 2 ((A V B V) → ⟨A, B⟩ = {{A}, {A, B}})
2 snexgOLD 3926 . . . . 5 (A V → {A} V)
32adantr 261 . . . 4 ((A V B V) → {A} V)
4 prexgOLD 3937 . . . 4 ((A V B V) → {A, B} V)
53, 4jca 290 . . 3 ((A V B V) → ({A} V {A, B} V))
6 prexgOLD 3937 . . 3 (({A} V {A, B} V) → {{A}, {A, B}} V)
75, 6syl 14 . 2 ((A V B V) → {{A}, {A, B}} V)
81, 7eqeltrd 2111 1 ((A V B V) → ⟨A, B V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
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-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376
This theorem is referenced by:  otth2  3969  opeliunxp  4338  opbrop  4362  relsnop  4387  op2ndb  4747  opswapg  4750  elxp4  4751  elxp5  4752  fvsn  5301  resfunexg  5325  fliftel  5376
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