MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dftr2 Structured version   Visualization version   GIF version

Theorem dftr2 5268
Description: An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. Using dftr2c 5269 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 3969 . 2 ( 𝐴𝐴 ↔ ∀𝑥(𝑥 𝐴𝑥𝐴))
2 df-tr 5267 . 2 (Tr 𝐴 𝐴𝐴)
3 19.23v 1946 . . . 4 (∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) → 𝑥𝐴))
4 eluni 4912 . . . . 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 3949   cuni 4909  Tr wtr 5266
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 3956  df-ss 3966  df-uni 4910  df-tr 5267
This theorem is referenced by:  dftr2c  5269  dftr5OLD  5271  trel  5275  ordelord  6387  suctr  6451  trom  7864  hartogs  9539  card2on  9549  trcl  9723  tskwe  9945  ondomon  10558  nosupno  27206  noinfno  27221  dftr6  34721  elpotr  34753  hftr  35154  dford4  41768  mnutrd  43039  tratrb  43297  trsbc  43301  truniALT  43302  sspwtr  43582  sspwtrALT  43583  sspwtrALT2  43584  pwtrVD  43585  pwtrrVD  43586  suctrALT  43587  suctrALT2VD  43597  suctrALT2  43598  tratrbVD  43622  trsbcVD  43638  truniALTVD  43639  trintALTVD  43641  trintALT  43642  suctrALTcf  43683  suctrALTcfVD  43684  suctrALT3  43685
  Copyright terms: Public domain W3C validator