Theorem dftr3 3879

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 A <-> A. x e. A x C_ A )

Distinct variable group:   x, A

Proof of Theorem dftr3

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr5 3878	. 2 |- ( Tr A <-> A. x e. A A. y e. x y e. A )
2		dfss3 2989	. . 3 |- ( x C_ A <-> A. y e. x y e. A )
3	2	ralbii 2372	. 2 |- ( A. x e. A x C_ A <-> A. x e. A A. y e. x y e. A )
4	1, 3	bitr4i 185	1 |- ( Tr A <-> A. x e. A x C_ A )

Syntax hints:   <-> wb 103   e. wcel 1433   A.wral 2348   C_ wss 2973   Tr wtr 3875

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-tr 3876

This theorem is referenced by:  trss  3884  trin  3885  triun  3888  trint  3890  tron  4137  ssorduni  4231  bj-nntrans2  10747  bj-omtrans2  10752