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Theorem dftr2 5214
Description: An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. Using dftr2c 5215 instead may avoid dependences on ax-11 2194. (Contributed by NM, 24-Apr-1994.)
Assertion
Ref Expression
dftr2 (Tr 𝐴 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dftr2
StepHypRef Expression
1 df-ss 3924 . 2 ( 𝐴𝐴 ↔ ∀𝑥(𝑥 𝐴𝑥𝐴))
2 df-tr 5213 . 2 (Tr 𝐴 𝐴𝐴)
3 19.23v 1965 . . . 4 (∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) → 𝑥𝐴))
4 eluni 4871 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
54imbi1i 352 . . . 4 ((𝑥 𝐴𝑥𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) → 𝑥𝐴))
63, 5bitr4i 281 . . 3 (∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ (𝑥 𝐴𝑥𝐴))
76albii 1842 . 2 (∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ ∀𝑥(𝑥 𝐴𝑥𝐴))
81, 2, 73bitr4i 306 1 (Tr 𝐴 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 400  wal 1561  wex 1802  wcel 2145  wss 3907   cuni 4868  Tr wtr 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737
This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-ss 3924  df-uni 4869  df-tr 5213
This theorem is referenced by:  dftr2c  5215  trel  5220  ordelord  6372  suctr  6438  trom  7859  hartogs  9494  card2on  9504  trcl  9685  tskwe  9924  ondomon  10535  nosupno  27825  noinfno  27840  bdayons  28427  dftr6  36114  elpotr  36142  hftr  36545  ttctr  36866  dfttc2g  36879  dfttc4lem2  36902  dford4  43618  mnutrd  44854  tratrb  45110  trsbc  45114  truniALT  45115  sspwtr  45394  sspwtrALT  45395  sspwtrALT2  45396  pwtrVD  45397  pwtrrVD  45398  suctrALT  45399  suctrALT2VD  45409  suctrALT2  45410  tratrbVD  45434  trsbcVD  45450  truniALTVD  45451  trintALTVD  45453  trintALT  45454  suctrALTcf  45495  suctrALTcfVD  45496  suctrALT3  45497
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