Theorem scott0f 30825

Description: A version of scott0 8321 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 8321	. 2 |- ( A = (/) <-> { w e. A | A. z e. A ( rank ` w ) C_ ( rank ` z ) } = (/) )
2		scott0f.1	. . . . . . 7 |- F/_ y A
3		nfcv 2619	. . . . . . 7 |- F/_ z A
4		nfv 1708	. . . . . . 7 |- F/ z ( rank ` x ) C_ ( rank ` y )
5		nfv 1708	. . . . . . 7 |- F/ y ( rank ` x ) C_ ( rank ` z )
6		fveq2 5872	. . . . . . . 8 |- ( y = z -> ( rank ` y ) = ( rank ` z ) )
7	6	sseq2d 3527	. . . . . . 7 |- ( y = z -> ( ( rank ` x ) C_ ( rank ` y ) <-> ( rank ` x ) C_ ( rank ` z ) ) )
8	2, 3, 4, 5, 7	cbvralf 3078	. . . . . 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 3098	. . . 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 2619	. . . . 5 |- F/_ w A
12		scott0f.2	. . . . 5 |- F/_ x A
13		nfv 1708	. . . . . 6 |- F/ x ( rank ` w ) C_ ( rank ` z )
14	12, 13	nfral 2843	. . . . 5 |- F/ x A. z e. A ( rank ` w ) C_ ( rank ` z )
15		nfv 1708	. . . . 5 |- F/ w A. z e. A ( rank ` x ) C_ ( rank ` z )
16		fveq2 5872	. . . . . . 7 |- ( w = x -> ( rank ` w ) = ( rank ` x ) )
17	16	sseq1d 3526	. . . . . 6 |- ( w = x -> ( ( rank ` w ) C_ ( rank ` z ) <-> ( rank ` x ) C_ ( rank ` z ) ) )
18	17	ralbidv 2896	. . . . 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 3107	. . . 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 2489	. . 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 2464	. 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 252	1 |- ( A = (/) <-> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = (/) )

Syntax hints:   <-> wb 184   = wceq 1395   e. wcel 1819   F/_wnfc 2605   A.wral 2807   {crab 2811   C_ wss 3471   (/)c0 3793   ` cfv 5594   rankcrnk 8198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-recs 7060  df-rdg 7094  df-r1 8199  df-rank 8200

This theorem is referenced by:  scottn0f  30826