Theorem elfpw 7206

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 3401	. 2 |- ( A e. ( ~P B i^i Fin ) <-> ( A e. ~P B /\ A e. Fin ) )
2		elpwg 3673	. . 3 |- ( A e. Fin -> ( A e. ~P B <-> A C_ B ) )
3	2	pm5.32ri 455	. 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 2203   i^i cin 3209   C_ wss 3210   ~Pcpw 3665   Fincfn 6966

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-in 3216  df-ss 3223  df-pw 3667

This theorem is referenced by: (None)