Theorem dftr4 2685

Description: An alternate way of defining a transitive class. Definition of [Enderton] p. 71.

Assertion

Ref	Expression
dftr4	|- (Tr A <-> A (_ P~A)

Proof of Theorem dftr4

Step	Hyp	Ref	Expression
1		visset 1813	. . . 4 |- x e. V
2	1	elpw 2404	. . 3 |- (x e. P~A <-> x (_ A)
3	2	ralbii 1667	. 2 |- (A.x e. A x e. P~A <-> A.x e. A x (_ A)
4		dfss3 2059	. 2 |- (A (_ P~A <-> A.x e. A x e. P~A)
5		dftr3 2684	. 2 |- (Tr A <-> A.x e. A x (_ A)
6	3, 4, 5	3bitr4r 184	1 |- (Tr A <-> A (_ P~A)

Syntax hints:   <-> wb 146   e. wcel 958  A.wral 1645   (_ wss 2047  P~cpw 2401  Tr wtr 2680

This theorem is referenced by:  tr0 2691  r1tr 4654

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-pw 2402  df-uni 2504  df-tr 2681

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