Theorem pwtr 4141

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 4139	. . 3 |- U. ~P A = A
2	1	sseq1i 3123	. 2 |- ( U. ~P A C_ ~P A <-> A C_ ~P A )
3		df-tr 4027	. 2 |- ( Tr ~P A <-> U. ~P A C_ ~P A )
4		dftr4 4031	. 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 3071   ~Pcpw 3510   U.cuni 3736   Tr wtr 4026

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-uni 3737  df-tr 4027

This theorem is referenced by: (None)