Theorem pwtr 4037

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 4035	. . 3 |- U. ~P A = A
2	1	sseq1i 3048	. 2 |- ( U. ~P A C_ ~P A <-> A C_ ~P A )
3		df-tr 3929	. 2 |- ( Tr ~P A <-> U. ~P A C_ ~P A )
4		dftr4 3933	. 2 |- ( Tr A <-> A C_ ~P A )
5	2, 3, 4	3bitr4ri 211	1 |- ( Tr A <-> Tr ~P A )

Syntax hints:   <-> wb 103   C_ wss 2997   ~Pcpw 3425   U.cuni 3648   Tr wtr 3928

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-uni 3649  df-tr 3929

This theorem is referenced by: (None)