Theorem dftr4 4197

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 4193	. 2 |- ( Tr A <-> U. A C_ A )
2		sspwuni 4060	. 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 3201   ~Pcpw 3656   U.cuni 3898   Tr wtr 4192

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-in 3207  df-ss 3214  df-pw 3658  df-uni 3899  df-tr 4193

This theorem is referenced by:  tr0  4203  pwtr  4317  pw1on  7487