Theorem pwtr 3955

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 3953	. . 3 |- U. ~P A = A
2	1	sseq1i 2969	. 2 |- ( U. ~P A C_ ~P A <-> A C_ ~P A )
3		df-tr 3855	. 2 |- ( Tr ~P A <-> U. ~P A C_ ~P A )
4		dftr4 3859	. 2 |- ( Tr A <-> A C_ ~P A )
5	2, 3, 4	3bitr4ri 202	1 |- ( Tr A <-> Tr ~P A )

Syntax hints:   <-> wb 98   C_ wss 2917   ~Pcpw 3359   U.cuni 3580   Tr wtr 3854

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-uni 3581  df-tr 3855

This theorem is referenced by: (None)