Theorem dftr4 4146

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 4142	. 2 |- ( Tr A <-> U. A C_ A )
2		sspwuni 4011	. 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 3165   ~Pcpw 3615   U.cuni 3849   Tr wtr 4141

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-in 3171  df-ss 3178  df-pw 3617  df-uni 3850  df-tr 4142

This theorem is referenced by:  tr0  4152  pwtr  4262  pw1on  7337