Theorem scott0s 8751

Description: Theorem scheme version of scott0 8749. The collection of all x of minimum rank such that ph ( x ) is true, is not empty iff there is an x such that ph ( x ) holds. (Contributed by NM, 13-Oct-2003.)

Assertion

Ref	Expression
scott0s	|- ( E. x ph <-> { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) } =/= (/) )

Distinct variable groups:   x, y   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem scott0s

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		abn0 3954	. 2 |- ( { x | ph } =/= (/) <-> E. x ph )
2		scott0 8749	. . . 4 |- ( { x | ph } = (/) <-> { z e. { x | ph } | A. y e. { x | ph } ( rank ` z ) C_ ( rank ` y ) } = (/) )
3		nfcv 2764	. . . . . . 7 |- F/_ z { x | ph }
4		nfab1 2766	. . . . . . 7 |- F/_ x { x | ph }
5		nfv 1843	. . . . . . . 8 |- F/ x ( rank ` z ) C_ ( rank ` y )
6	4, 5	nfral 2945	. . . . . . 7 |- F/ x A. y e. { x | ph } ( rank ` z ) C_ ( rank ` y )
7		nfv 1843	. . . . . . 7 |- F/ z A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y )
8		fveq2 6191	. . . . . . . . 9 |- ( z = x -> ( rank ` z ) = ( rank ` x ) )
9	8	sseq1d 3632	. . . . . . . 8 |- ( z = x -> ( ( rank ` z ) C_ ( rank ` y ) <-> ( rank ` x ) C_ ( rank ` y ) ) )
10	9	ralbidv 2986	. . . . . . 7 |- ( z = x -> ( A. y e. { x | ph } ( rank ` z ) C_ ( rank ` y ) <-> A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) ) )
11	3, 4, 6, 7, 10	cbvrab 3198	. . . . . 6 |- { z e. { x | ph } | A. y e. { x | ph } ( rank ` z ) C_ ( rank ` y ) } = { x e. { x | ph } | A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) }
12		df-rab 2921	. . . . . 6 |- { x e. { x | ph } | A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) } = { x | ( x e. { x | ph } /\ A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) ) }
13		abid 2610	. . . . . . . 8 |- ( x e. { x | ph } <-> ph )
14		df-ral 2917	. . . . . . . . 9 |- ( A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) <-> A. y ( y e. { x | ph } -> ( rank ` x ) C_ ( rank ` y ) ) )
15		df-sbc 3436	. . . . . . . . . . 11 |- ( [. y / x ]. ph <-> y e. { x | ph } )
16	15	imbi1i 339	. . . . . . . . . 10 |- ( ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) <-> ( y e. { x | ph } -> ( rank ` x ) C_ ( rank ` y ) ) )
17	16	albii 1747	. . . . . . . . 9 |- ( A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) <-> A. y ( y e. { x | ph } -> ( rank ` x ) C_ ( rank ` y ) ) )
18	14, 17	bitr4i 267	. . . . . . . 8 |- ( A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) <-> A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) )
19	13, 18	anbi12i 733	. . . . . . 7 |- ( ( x e. { x | ph } /\ A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) ) <-> ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) )
20	19	abbii 2739	. . . . . 6 |- { x | ( x e. { x | ph } /\ A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) ) } = { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) }
21	11, 12, 20	3eqtri 2648	. . . . 5 |- { z e. { x | ph } | A. y e. { x | ph } ( rank ` z ) C_ ( rank ` y ) } = { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) }
22	21	eqeq1i 2627	. . . 4 |- ( { z e. { x | ph } | A. y e. { x | ph } ( rank ` z ) C_ ( rank ` y ) } = (/) <-> { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) } = (/) )
23	2, 22	bitri 264	. . 3 |- ( { x | ph } = (/) <-> { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) } = (/) )
24	23	necon3bii 2846	. 2 |- ( { x | ph } =/= (/) <-> { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) } =/= (/) )
25	1, 24	bitr3i 266	1 |- ( E. x ph <-> { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) } =/= (/) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   {crab 2916   [.wsbc 3435   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:  hta  8760