Theorem dftr3 4145

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 4144	. 2 |- ( Tr A <-> A. x e. A A. y e. x y e. A )
2		dfss3 3181	. . 3 |- ( x C_ A <-> A. y e. x y e. A )
3	2	ralbii 2511	. 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 2175   A.wral 2483   C_ wss 3165   Tr wtr 4141

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-in 3171  df-ss 3178  df-uni 3850  df-tr 4142

This theorem is referenced by:  trss  4150  trin  4151  triun  4154  trint  4156  tron  4427  ssorduni  4533  pw1on  7320  bj-nntrans2  15752  bj-omtrans2  15757