Theorem dftr4 4136

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 4132	. 2 |- ( Tr A <-> U. A C_ A )
2		sspwuni 4001	. 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 3157   ~Pcpw 3605   U.cuni 3839   Tr wtr 4131

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-pw 3607  df-uni 3840  df-tr 4132

This theorem is referenced by:  tr0  4142  pwtr  4252  pw1on  7293