Theorem pwtr 4197

Description: A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)

Assertion

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

Proof of Theorem pwtr

Step	Hyp	Ref	Expression
1		unipw 4195	. . 3 |- U. ~P A = A
2	1	sseq1i 3168	. 2 |- ( U. ~P A C_ ~P A <-> A C_ ~P A )
3		df-tr 4081	. 2 |- ( Tr ~P A <-> U. ~P A C_ ~P A )
4		dftr4 4085	. 2 |- ( Tr A <-> A C_ ~P A )
5	2, 3, 4	3bitr4ri 212	1 |- ( Tr A <-> Tr ~P A )

Syntax hints:   <-> wb 104   C_ wss 3116   ~Pcpw 3559   U.cuni 3789   Tr wtr 4080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-uni 3790  df-tr 4081

This theorem is referenced by: (None)