Theorem dftr3 4212

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 4211	. 2 |- ( Tr A <-> A. x e. A A. y e. x y e. A )
2		dfss3 3227	. . 3 |- ( x C_ A <-> A. y e. x y e. A )
3	2	ralbii 2548	. 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 2203   A.wral 2520   C_ wss 3211   Tr wtr 4208

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815  df-in 3217  df-ss 3224  df-uni 3915  df-tr 4209

This theorem is referenced by:  trss  4217  trin  4218  triun  4221  trint  4223  tron  4503  ssorduni  4609  pw1on  7536  bj-nntrans2  16722  bj-omtrans2  16727