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Theorem dftr3 3849
Description: An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.)
Assertion
Ref Expression
dftr3 (Tr Ax A xA)
Distinct variable group:   x,A

Proof of Theorem dftr3
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dftr5 3848 . 2 (Tr Ax A y x y A)
2 dfss3 2929 . . 3 (xAy x y A)
32ralbii 2324 . 2 (x A xAx A y x y A)
41, 3bitr4i 176 1 (Tr Ax A xA)
Colors of variables: wff set class
Syntax hints:  wb 98   wcel 1390  wral 2300  wss 2911  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-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846
This theorem is referenced by:  trss  3854  trin  3855  triun  3858  trint  3860  tron  4085  ssorduni  4179  bj-nntrans2  9386  bj-omtrans2  9391
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