Theorem pwtr 4305

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 4303	. . 3 |- U. ~P A = A
2	1	sseq1i 3250	. 2 |- ( U. ~P A C_ ~P A <-> A C_ ~P A )
3		df-tr 4183	. 2 |- ( Tr ~P A <-> U. ~P A C_ ~P A )
4		dftr4 4187	. 2 |- ( Tr A <-> A C_ ~P A )
5	2, 3, 4	3bitr4ri 213	1 |- ( Tr A <-> Tr ~P A )

Syntax hints:   <-> wb 105   C_ wss 3197   ~Pcpw 3649   U.cuni 3888   Tr wtr 4182

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-uni 3889  df-tr 4183

This theorem is referenced by: (None)