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Theorem dftr2 5165
Description: An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (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 3953 . 2 ( 𝐴𝐴 ↔ ∀𝑥(𝑥 𝐴𝑥𝐴))
2 df-tr 5164 . 2 (Tr 𝐴 𝐴𝐴)
3 19.23v 1936 . . . 4 (∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) → 𝑥𝐴))
4 eluni 4833 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
54imbi1i 352 . . . 4 ((𝑥 𝐴𝑥𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) → 𝑥𝐴))
63, 5bitr4i 280 . . 3 (∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ (𝑥 𝐴𝑥𝐴))
76albii 1813 . 2 (∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ ∀𝑥(𝑥 𝐴𝑥𝐴))
81, 2, 73bitr4i 305 1 (Tr 𝐴 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1528  wex 1773  wcel 2107  wss 3934   cuni 4830  Tr wtr 5163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-in 3941  df-ss 3950  df-uni 4831  df-tr 5164
This theorem is referenced by:  dftr5  5166  trel  5170  ordelord  6206  suctr  6267  ordom  7581  hartogs  9000  card2on  9010  trcl  9162  tskwe  9371  ondomon  9977  dftr6  32979  elpotr  33019  nosupno  33196  hftr  33636  dford4  39616  mnutrd  40606  tratrb  40860  trsbc  40864  truniALT  40865  sspwtr  41145  sspwtrALT  41146  sspwtrALT2  41147  pwtrVD  41148  pwtrrVD  41149  suctrALT  41150  suctrALT2VD  41160  suctrALT2  41161  tratrbVD  41185  trsbcVD  41201  truniALTVD  41202  trintALTVD  41204  trintALT  41205  suctrALTcf  41246  suctrALTcfVD  41247  suctrALT3  41248
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