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Theorem dftr2 5213
Description: An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. Using dftr2c 5214 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 5212 . 2 (Tr 𝐴 𝐴𝐴)
3 19.23v 1965 . . . 4 (∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) → 𝑥𝐴))
4 eluni 4870 . . . . 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 4867  Tr wtr 5211
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 4868  df-tr 5212
This theorem is referenced by:  dftr2c  5214  trel  5219  ordelord  6371  suctr  6438  trom  7859  hartogs  9494  card2on  9504  trcl  9685  tskwe  9924  ondomon  10535  nosupno  27821  noinfno  27836  bdayons  28423  dftr6  36109  elpotr  36137  hftr  36540  ttctr  36861  dfttc2g  36874  dfttc4lem2  36897  dford4  43613  mnutrd  44849  tratrb  45104  trsbc  45108  truniALT  45109  sspwtr  45388  sspwtrALT  45389  sspwtrALT2  45390  pwtrVD  45391  pwtrrVD  45392  suctrALT  45393  suctrALT2VD  45403  suctrALT2  45404  tratrbVD  45428  trsbcVD  45444  truniALTVD  45445  trintALTVD  45447  trintALT  45448  suctrALTcf  45489  suctrALTcfVD  45490  suctrALT3  45491
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