Theorem dftr3 4030

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 4029	. 2 |- ( Tr A <-> A. x e. A A. y e. x y e. A )
2		dfss3 3087	. . 3 |- ( x C_ A <-> A. y e. x y e. A )
3	2	ralbii 2441	. 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 1480   A.wral 2416   C_ wss 3071   Tr wtr 4026

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027

This theorem is referenced by:  trss  4035  trin  4036  triun  4039  trint  4041  tron  4304  ssorduni  4403  bj-nntrans2  13150  bj-omtrans2  13155