Theorem elfpw 7255

Description: Membership in a class of finite subsets. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
elfpw	|- ( A e. ( ~P B i^i Fin ) <-> ( A C_ B /\ A e. Fin ) )

Proof of Theorem elfpw

Step	Hyp	Ref	Expression
1		elin 3412	. 2 |- ( A e. ( ~P B i^i Fin ) <-> ( A e. ~P B /\ A e. Fin ) )
2		elpwg 3696	. . 3 |- ( A e. Fin -> ( A e. ~P B <-> A C_ B ) )
3	2	pm5.32ri 459	. 2 |- ( ( A e. ~P B /\ A e. Fin ) <-> ( A C_ B /\ A e. Fin ) )
4	1, 3	bitri 184	1 |- ( A e. ( ~P B i^i Fin ) <-> ( A C_ B /\ A e. Fin ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2209   i^i cin 3219   C_ wss 3220   ~Pcpw 3688   Fincfn 7015

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-ext 2220

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-v 2823  df-in 3226  df-ss 3233  df-pw 3690

This theorem is referenced by:  ballotfilemelo  13203