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Theorem dftr2 3847
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 Axy((x y y A) → x A))
Distinct variable group:   x,y,A

Proof of Theorem dftr2
StepHypRef Expression
1 dfss2 2928 . 2 ( AAx(x Ax A))
2 df-tr 3846 . 2 (Tr A AA)
3 19.23v 1760 . . . 4 (y((x y y A) → x A) ↔ (y(x y y A) → x A))
4 eluni 3574 . . . . 5 (x Ay(x y y A))
54imbi1i 227 . . . 4 ((x Ax A) ↔ (y(x y y A) → x A))
63, 5bitr4i 176 . . 3 (y((x y y A) → x A) ↔ (x Ax A))
76albii 1356 . 2 (xy((x y y A) → x A) ↔ x(x Ax A))
81, 2, 73bitr4i 201 1 (Tr Axy((x y y A) → x A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wex 1378   wcel 1390  wss 2911   cuni 3571  Tr wtr 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846
This theorem is referenced by:  dftr5  3848  trel  3852  suctr  4124  ordtriexmidlem  4208  onsucelsucexmidlem  4214  ordsuc  4241  tfi  4248  ordom  4272
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