Theorem scott0f 33977

Description: A version of scott0 8749 with non-free variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)

Hypotheses

Ref	Expression
scott0f.1	|- F/_ y A
scott0f.2	|- F/_ x A

Assertion

Ref	Expression
scott0f	|- ( A = (/) <-> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = (/) )

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)

Proof of Theorem scott0f

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scott0 8749	. 2 |- ( A = (/) <-> { w e. A | A. z e. A ( rank ` w ) C_ ( rank ` z ) } = (/) )
2		scott0f.1	. . . . . . 7 |- F/_ y A
3		nfcv 2764	. . . . . . 7 |- F/_ z A
4		nfv 1843	. . . . . . 7 |- F/ z ( rank ` x ) C_ ( rank ` y )
5		nfv 1843	. . . . . . 7 |- F/ y ( rank ` x ) C_ ( rank ` z )
6		fveq2 6191	. . . . . . . 8 |- ( y = z -> ( rank ` y ) = ( rank ` z ) )
7	6	sseq2d 3633	. . . . . . 7 |- ( y = z -> ( ( rank ` x ) C_ ( rank ` y ) <-> ( rank ` x ) C_ ( rank ` z ) ) )
8	2, 3, 4, 5, 7	cbvralf 3165	. . . . . 6 |- ( A. y e. A ( rank ` x ) C_ ( rank ` y ) <-> A. z e. A ( rank ` x ) C_ ( rank ` z ) )
9	8	a1i 11	. . . . 5 |- ( x e. A -> ( A. y e. A ( rank ` x ) C_ ( rank ` y ) <-> A. z e. A ( rank ` x ) C_ ( rank ` z ) ) )
10	9	rabbiia 3185	. . . 4 |- { 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 2764	. . . . 5 |- F/_ w A
12		scott0f.2	. . . . 5 |- F/_ x A
13		nfv 1843	. . . . . 6 |- F/ x ( rank ` w ) C_ ( rank ` z )
14	12, 13	nfral 2945	. . . . 5 |- F/ x A. z e. A ( rank ` w ) C_ ( rank ` z )
15		nfv 1843	. . . . 5 |- F/ w A. z e. A ( rank ` x ) C_ ( rank ` z )
16		fveq2 6191	. . . . . . 7 |- ( w = x -> ( rank ` w ) = ( rank ` x ) )
17	16	sseq1d 3632	. . . . . 6 |- ( w = x -> ( ( rank ` w ) C_ ( rank ` z ) <-> ( rank ` x ) C_ ( rank ` z ) ) )
18	17	ralbidv 2986	. . . . 5 |- ( w = x -> ( A. z e. A ( rank ` w ) C_ ( rank ` z ) <-> A. z e. A ( rank ` x ) C_ ( rank ` z ) ) )
19	11, 12, 14, 15, 18	cbvrab 3198	. . . 4 |- { w e. A | A. z e. A ( rank ` w ) C_ ( rank ` z ) } = { x e. A | A. z e. A ( rank ` x ) C_ ( rank ` z ) }
20	10, 19	eqtr4i 2647	. . 3 |- { 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	20	eqeq1i 2627	. 2 |- ( { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = (/) <-> { w e. A | A. z e. A ( rank ` w ) C_ ( rank ` z ) } = (/) )
22	1, 21	bitr4i 267	1 |- ( A = (/) <-> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = (/) )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   F/_wnfc 2751   A.wral 2912   {crab 2916   C_ wss 3574   (/)c0 3915   ` cfv 5888   rankcrnk 8626

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628

This theorem is referenced by:  scottn0f  33978