Theorem pwtr 4354

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 4352	. . 3 |- U. ~P A = A
2	1	sseq1i 3274	. 2 |- ( U. ~P A C_ ~P A <-> A C_ ~P A )
3		df-tr 4225	. 2 |- ( Tr ~P A <-> U. ~P A C_ ~P A )
4		dftr4 4229	. 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 3220   ~Pcpw 3685   U.cuni 3930   Tr wtr 4224

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-uni 3931  df-tr 4225

This theorem is referenced by: (None)