Theorem dftr4 2690

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 1816	. . . 4 |- x e. V
2	1	elpw 2408	. . 3 |- (x e. P~A <-> x (_ A)
3	2	ralbii 1670	. 2 |- (A.x e. A x e. P~A <-> A.x e. A x (_ A)
4		dfss3 2062	. 2 |- (A (_ P~A <-> A.x e. A x e. P~A)
5		dftr3 2689	. 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 960  A.wral 1648   (_ wss 2050  P~cpw 2405  Tr wtr 2685

This theorem is referenced by:  tr0 2696  r1tr 4664

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-in 2054  df-ss 2056  df-pw 2406  df-uni 2508  df-tr 2686

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