Theorem isfin2 9116

Description: Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)

Assertion

Ref	Expression
isfin2	|- ( A e. V -> ( A e. FinII <-> A. y e. ~P ~P A ( ( y =/= (/) /\ [ C.] Or y ) -> U. y e. y ) ) )

Distinct variable group:   y, A

Allowed substitution hint:   V( y)

Proof of Theorem isfin2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4161	. . . 4 |- ( x = A -> ~P x = ~P A )
2	1	pweqd 4163	. . 3 |- ( x = A -> ~P ~P x = ~P ~P A )
3	2	raleqdv 3144	. 2 |- ( x = A -> ( A. y e. ~P ~P x ( ( y =/= (/) /\ [ C.] Or y ) -> U. y e. y ) <-> A. y e. ~P ~P A ( ( y =/= (/) /\ [ C.] Or y ) -> U. y e. y ) ) )
4		df-fin2 9108	. 2 |- FinII = { x | A. y e. ~P ~P x ( ( y =/= (/) /\ [ C.] Or y ) -> U. y e. y ) }
5	3, 4	elab2g 3353	1 |- ( A e. V -> ( A e. FinII <-> A. y e. ~P ~P A ( ( y =/= (/) /\ [ C.] Or y ) -> U. y e. y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   Or wor 5034   [ C.] crpss 6936  Fin^IIcfin2 9101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-fin2 9108

This theorem is referenced by:  fin2i  9117  isfin2-2  9141  ssfin2  9142  enfin2i  9143  fin12  9235  fin1a2s  9236