Theorem dftr3 4084

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 4083	. 2 |- ( Tr A <-> A. x e. A A. y e. x y e. A )
2		dfss3 3132	. . 3 |- ( x C_ A <-> A. y e. x y e. A )
3	2	ralbii 2472	. 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 2136   A.wral 2444   C_ wss 3116   Tr wtr 4080

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081

This theorem is referenced by:  trss  4089  trin  4090  triun  4093  trint  4095  tron  4360  ssorduni  4464  pw1on  7182  bj-nntrans2  13834  bj-omtrans2  13839