Theorem pwtr 4252

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 4250	. . 3 |- U. ~P A = A
2	1	sseq1i 3209	. 2 |- ( U. ~P A C_ ~P A <-> A C_ ~P A )
3		df-tr 4132	. 2 |- ( Tr ~P A <-> U. ~P A C_ ~P A )
4		dftr4 4136	. 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 3157   ~Pcpw 3605   U.cuni 3839   Tr wtr 4131

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-uni 3840  df-tr 4132

This theorem is referenced by: (None)