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Theorem opexg 3916
Description: An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.)
Assertion
Ref Expression
opexg ((A 𝑉 B 𝑊) → ⟨A, B V)

Proof of Theorem opexg
StepHypRef Expression
1 dfopg 3499 . 2 ((A 𝑉 B 𝑊) → ⟨A, B⟩ = {{A}, {A, B}})
2 elex 2541 . . . . 5 (A 𝑉A V)
3 snexgOLD 3887 . . . . 5 (A V → {A} V)
42, 3syl 14 . . . 4 (A 𝑉 → {A} V)
54adantr 261 . . 3 ((A 𝑉 B 𝑊) → {A} V)
6 elex 2541 . . . 4 (B 𝑊B V)
7 prexgOLD 3898 . . . 4 ((A V B V) → {A, B} V)
82, 6, 7syl2an 273 . . 3 ((A 𝑉 B 𝑊) → {A, B} V)
9 prexgOLD 3898 . . 3 (({A} V {A, B} V) → {{A}, {A, B}} V)
105, 8, 9syl2anc 393 . 2 ((A 𝑉 B 𝑊) → {{A}, {A, B}} V)
111, 10eqeltrd 2096 1 ((A 𝑉 B 𝑊) → ⟨A, B V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1375  Vcvv 2533  {csn 3327  {cpr 3328  cop 3330
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 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2535  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336
This theorem is referenced by:  opex  3918  fliftel1  5326  oprabid  5430  eloprabga  5483  op1st  5662  op2nd  5663  ot1stg  5668  ot2ndg  5669  ot3rdgg  5670  elxp6  5685  mpt2fvex  5718  algrflem  5739  mpt2xopoveq  5743  brtposg  5757  tfrlemisucaccv  5825  tfrlemibxssdm  5827  tfrlemibfn  5828  tfrlemi14d  5833
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