Theorem findcard2s

Description: Variation of findcard2 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012)

	Ref	Expression
Hypotheses	findcard2s.1	|- ( x = (/) -> ( ph <-> ps ) )
	findcard2s.2	|- ( x = y -> ( ph <-> ch ) )
	findcard2s.3	|- ( x = ( y u. { z } ) -> ( ph <-> th ) )
	findcard2s.4	|- ( x = A -> ( ph <-> ta ) )
	findcard2s.5	|- ps
	findcard2s.6	|- ( ( y e. Fin /\ -. z e. y ) -> ( ch -> th ) )
Assertion	findcard2s	|- ( A e. Fin -> ta )

Proof

Step	Hyp	Ref	Expression
1		findcard2s.1	|- ( x = (/) -> ( ph <-> ps ) )
2		findcard2s.2	|- ( x = y -> ( ph <-> ch ) )
3		findcard2s.3	|- ( x = ( y u. { z } ) -> ( ph <-> th ) )
4		findcard2s.4	|- ( x = A -> ( ph <-> ta ) )
5		findcard2s.5	|- ps
6		findcard2s.6	|- ( ( y e. Fin /\ -. z e. y ) -> ( ch -> th ) )
7	6	ex	|- ( y e. Fin -> ( -. z e. y -> ( ch -> th ) ) )
8		snssi	|- ( z e. y -> { z } C_ y )
9		ssequn1	|- ( { z } C_ y <-> ( { z } u. y ) = y )
10	8 9	sylib	|- ( z e. y -> ( { z } u. y ) = y )
11		uncom	|- ( { z } u. y ) = ( y u. { z } )
12	10 11	eqtr3di	|- ( z e. y -> y = ( y u. { z } ) )
13		vex	|- y e. _V
14	13	eqvinc	|- ( y = ( y u. { z } ) <-> E. x ( x = y /\ x = ( y u. { z } ) ) )
15	12 14	sylib	|- ( z e. y -> E. x ( x = y /\ x = ( y u. { z } ) ) )
16	2	bicomd	|- ( x = y -> ( ch <-> ph ) )
17	16 3	sylan9bb	|- ( ( x = y /\ x = ( y u. { z } ) ) -> ( ch <-> th ) )
18	17	exlimiv	|- ( E. x ( x = y /\ x = ( y u. { z } ) ) -> ( ch <-> th ) )
19	15 18	syl	|- ( z e. y -> ( ch <-> th ) )
20	19	biimpd	|- ( z e. y -> ( ch -> th ) )
21	7 20	pm2.61d2	|- ( y e. Fin -> ( ch -> th ) )
22	1 2 3 4 5 21	findcard2	|- ( A e. Fin -> ta )