Theorem dftr3 2045

Description: An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35.

Assertion

Ref	Expression
dftr3	|- (Tr A <-> A.x e. A x (_ A)

Distinct variable group(s):   x,A

Proof of Theorem dftr3

Step	Hyp	Ref	Expression
1		dftr5 2044	. 2 |- (Tr A <-> A.x e. A A.y e. x y e. A)
2		dfss3 1498	. . . 4 |- (x (_ A <-> A.y e. x y e. A)
3	2	biral 1223	. . 3 |- (A.x e. A x (_ A <-> A.x e. A A.y e. x y e. A)
4	3	bicomi 150	. 2 |- (A.x e. A A.y e. x y e. A <-> A.x e. A x (_ A)
5	1, 4	bitr 151	1 |- (Tr A <-> A.x e. A x (_ A)

Syntax hints:   <-> wb 127   e. wcel 1092  A.wral 1201   (_ wss 1487  Tr wtr 2041

This theorem is referenced by:  dftr4 2046  trss 2050  trin 2051  ordon 2238  ssorduni 2249  suceloni 2314  r1tr 3498

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-uni 1920  df-tr 2042

metamath.org