Theorem dftr3 4162

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 4161	. 2 |- ( Tr A <-> A. x e. A A. y e. x y e. A )
2		dfss3 3190	. . 3 |- ( x C_ A <-> A. y e. x y e. A )
3	2	ralbii 2514	. 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 2178   A.wral 2486   C_ wss 3174   Tr wtr 4158

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-in 3180  df-ss 3187  df-uni 3865  df-tr 4159

This theorem is referenced by:  trss  4167  trin  4168  triun  4171  trint  4173  tron  4447  ssorduni  4553  pw1on  7372  bj-nntrans2  16087  bj-omtrans2  16092