Theorem dftr3 4131

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 4130	. 2 |- ( Tr A <-> A. x e. A A. y e. x y e. A )
2		dfss3 3169	. . 3 |- ( x C_ A <-> A. y e. x y e. A )
3	2	ralbii 2500	. 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 2164   A.wral 2472   C_ wss 3153   Tr wtr 4127

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-in 3159  df-ss 3166  df-uni 3836  df-tr 4128

This theorem is referenced by:  trss  4136  trin  4137  triun  4140  trint  4142  tron  4413  ssorduni  4519  pw1on  7286  bj-nntrans2  15444  bj-omtrans2  15449