Theorem dftr3 4091

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 4090	. 2 |- ( Tr A <-> A. x e. A A. y e. x y e. A )
2		dfss3 3137	. . 3 |- ( x C_ A <-> A. y e. x y e. A )
3	2	ralbii 2476	. 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 2141   A.wral 2448   C_ wss 3121   Tr wtr 4087

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088

This theorem is referenced by:  trss  4096  trin  4097  triun  4100  trint  4102  tron  4367  ssorduni  4471  pw1on  7203  bj-nntrans2  13987  bj-omtrans2  13992