Theorem pwtr 4149

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 4147	. . 3 |- U. ~P A = A
2	1	sseq1i 3128	. 2 |- ( U. ~P A C_ ~P A <-> A C_ ~P A )
3		df-tr 4035	. 2 |- ( Tr ~P A <-> U. ~P A C_ ~P A )
4		dftr4 4039	. 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 3076   ~Pcpw 3515   U.cuni 3744   Tr wtr 4034

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-uni 3745  df-tr 4035

This theorem is referenced by: (None)