Theorem prsspwg 3523

Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.) (Revised by NM, 18-Jan-2018.)

Assertion

Ref	Expression
prsspwg	|- ( ( A e. V /\ B e. W ) -> ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) ) )

Proof of Theorem prsspwg

Step	Hyp	Ref	Expression
1		prssg 3521	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. ~P C /\ B e. ~P C ) <-> { A , B } C_ ~P C ) )
2		elpwg 3367	. . 3 |- ( A e. V -> ( A e. ~P C <-> A C_ C ) )
3		elpwg 3367	. . 3 |- ( B e. W -> ( B e. ~P C <-> B C_ C ) )
4	2, 3	bi2anan9 538	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. ~P C /\ B e. ~P C ) <-> ( A C_ C /\ B C_ C ) ) )
5	1, 4	bitr3d 179	1 |- ( ( A e. V /\ B e. W ) -> ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   e. wcel 1393   C_ wss 2917   ~Pcpw 3359   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382

This theorem is referenced by: (None)