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Theorem dftr2 3826
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 2907 . 2 ( AAx(x Ax A))
2 df-tr 3825 . 2 (Tr A AA)
3 19.23v 1741 . . . 4 (y((x y y A) → x A) ↔ (y(x y y A) → x A))
4 eluni 3553 . . . . 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 1335 . 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 1224  wex 1358   wcel 1370  wss 2890   cuni 3550  Tr wtr 3824
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-in 2897  df-ss 2904  df-uni 3551  df-tr 3825
This theorem is referenced by:  dftr5  3827  trel  3831  suctrALT  4103  suctr  4104  ordtriexmidlem  4188  onsucelsucexmidlem  4194  ordsuc  4221  tfi  4228  ordom  4252
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