Theorem pwtr 4204

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 4202	. . 3 |- U. ~P A = A
2	1	sseq1i 3173	. 2 |- ( U. ~P A C_ ~P A <-> A C_ ~P A )
3		df-tr 4088	. 2 |- ( Tr ~P A <-> U. ~P A C_ ~P A )
4		dftr4 4092	. 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 3121   ~Pcpw 3566   U.cuni 3796   Tr wtr 4087

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-uni 3797  df-tr 4088

This theorem is referenced by: (None)