Theorem scott0f

Description: A version of scott0 with nonfree variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018)

	Ref	Expression
Hypotheses	scott0f.1	|- F/_ y A
	scott0f.2	|- F/_ x A
Assertion	scott0f	|- ( A = (/) <-> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = (/) )

Proof

Step	Hyp	Ref	Expression
1		scott0f.1	|- F/_ y A
2		scott0f.2	|- F/_ x A
3		scott0	|- ( A = (/) <-> { w e. A | A. z e. A ( rank ` w ) C_ ( rank ` z ) } = (/) )
4		nfcv	|- F/_ z A
5		nfv	|- F/ z ( rank ` x ) C_ ( rank ` y )
6		nfv	|- F/ y ( rank ` x ) C_ ( rank ` z )
7		fveq2	|- ( y = z -> ( rank ` y ) = ( rank ` z ) )
8	7	sseq2d	|- ( y = z -> ( ( rank ` x ) C_ ( rank ` y ) <-> ( rank ` x ) C_ ( rank ` z ) ) )
9	1 4 5 6 8	cbvralfw	|- ( A. y e. A ( rank ` x ) C_ ( rank ` y ) <-> A. z e. A ( rank ` x ) C_ ( rank ` z ) )
10	9	rabbii	|- { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = { x e. A | A. z e. A ( rank ` x ) C_ ( rank ` z ) }
11		nfcv	|- F/_ w A
12		nfv	|- F/ x ( rank ` w ) C_ ( rank ` z )
13	2 12	nfralw	|- F/ x A. z e. A ( rank ` w ) C_ ( rank ` z )
14		nfv	|- F/ w A. z e. A ( rank ` x ) C_ ( rank ` z )
15		fveq2	|- ( w = x -> ( rank ` w ) = ( rank ` x ) )
16	15	sseq1d	|- ( w = x -> ( ( rank ` w ) C_ ( rank ` z ) <-> ( rank ` x ) C_ ( rank ` z ) ) )
17	16	ralbidv	|- ( w = x -> ( A. z e. A ( rank ` w ) C_ ( rank ` z ) <-> A. z e. A ( rank ` x ) C_ ( rank ` z ) ) )
18	11 2 13 14 17	cbvrabw	|- { w e. A | A. z e. A ( rank ` w ) C_ ( rank ` z ) } = { x e. A | A. z e. A ( rank ` x ) C_ ( rank ` z ) }
19	10 18	eqtr4i	|- { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = { w e. A | A. z e. A ( rank ` w ) C_ ( rank ` z ) }
20	19	eqeq1i	|- ( { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = (/) <-> { w e. A | A. z e. A ( rank ` w ) C_ ( rank ` z ) } = (/) )
21	3 20	bitr4i	|- ( A = (/) <-> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = (/) )