Theorem dftr3 4191

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 4190	. 2 |- ( Tr A <-> A. x e. A A. y e. x y e. A )
2		dfss3 3216	. . 3 |- ( x C_ A <-> A. y e. x y e. A )
3	2	ralbii 2538	. 2 |- ( A. x e. A x C_ A <-> A. x e. A A. y e. x y e. A )
4	1, 3	bitr4i 187	1 |- ( Tr A <-> A. x e. A x C_ A )

Syntax hints:   <-> wb 105   e. wcel 2202   A.wral 2510   C_ wss 3200   Tr wtr 4187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188

This theorem is referenced by:  trss  4196  trin  4197  triun  4200  trint  4202  tron  4479  ssorduni  4585  pw1on  7443  bj-nntrans2  16547  bj-omtrans2  16552