Theorem dftr4 4121

Description: An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)

Assertion

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

Proof of Theorem dftr4

Step	Hyp	Ref	Expression
1		df-tr 4117	. 2 |- ( Tr A <-> U. A C_ A )
2		sspwuni 3986	. 2 |- ( A C_ ~P A <-> U. A C_ A )
3	1, 2	bitr4i 187	1 |- ( Tr A <-> A C_ ~P A )

Syntax hints:   <-> wb 105   C_ wss 3144   ~Pcpw 3590   U.cuni 3824   Tr wtr 4116

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-uni 3825  df-tr 4117

This theorem is referenced by:  tr0  4127  pwtr  4237  pw1on  7256