Theorem dftr3 4038

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 4037	. 2 |- ( Tr A <-> A. x e. A A. y e. x y e. A )
2		dfss3 3092	. . 3 |- ( x C_ A <-> A. y e. x y e. A )
3	2	ralbii 2444	. 2 |- ( A. x e. A x C_ A <-> A. x e. A A. y e. x y e. A )
4	1, 3	bitr4i 186	1 |- ( Tr A <-> A. x e. A x C_ A )

Syntax hints:   <-> wb 104   e. wcel 1481   A.wral 2417   C_ wss 3076   Tr wtr 4034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-in 3082  df-ss 3089  df-uni 3745  df-tr 4035

This theorem is referenced by:  trss  4043  trin  4044  triun  4047  trint  4049  tron  4312  ssorduni  4411  bj-nntrans2  13321  bj-omtrans2  13326