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Theorem dftr2 5267
Description: An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. Using dftr2c 5268 instead may avoid dependences on ax-11 2155. (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 dfss2 3968 . 2 ( 𝐴𝐴 ↔ ∀𝑥(𝑥 𝐴𝑥𝐴))
2 df-tr 5266 . 2 (Tr 𝐴 𝐴𝐴)
3 19.23v 1946 . . . 4 (∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) → 𝑥𝐴))
4 eluni 4911 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
54imbi1i 350 . . . 4 ((𝑥 𝐴𝑥𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) → 𝑥𝐴))
63, 5bitr4i 278 . . 3 (∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ (𝑥 𝐴𝑥𝐴))
76albii 1822 . 2 (∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ ∀𝑥(𝑥 𝐴𝑥𝐴))
81, 2, 73bitr4i 303 1 (Tr 𝐴 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wex 1782  wcel 2107  wss 3948   cuni 4908  Tr wtr 5265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3955  df-ss 3965  df-uni 4909  df-tr 5266
This theorem is referenced by:  dftr2c  5268  dftr5OLD  5270  trel  5274  ordelord  6384  suctr  6448  trom  7861  hartogs  9536  card2on  9546  trcl  9720  tskwe  9942  ondomon  10555  nosupno  27196  noinfno  27211  dftr6  34710  elpotr  34742  hftr  35143  dford4  41754  mnutrd  43025  tratrb  43283  trsbc  43287  truniALT  43288  sspwtr  43568  sspwtrALT  43569  sspwtrALT2  43570  pwtrVD  43571  pwtrrVD  43572  suctrALT  43573  suctrALT2VD  43583  suctrALT2  43584  tratrbVD  43608  trsbcVD  43624  truniALTVD  43625  trintALTVD  43627  trintALT  43628  suctrALTcf  43669  suctrALTcfVD  43670  suctrALT3  43671
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