Theorem frmin 31739

Description: Every (possibly proper) subclass of a class A with a founded, set-like relation R has a minimal element. Lemma 4.3 of Don Monk's notes for Advanced Set Theory, which can be found at http://euclid.colorado.edu/~monkd/settheory. This is a very strong generalization of tz6.26 5711 and tz7.5 5744. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Assertion

Ref	Expression
frmin	|- ( ( ( R Fr A /\ R Se A ) /\ ( B C_ A /\ B =/= (/) ) ) -> E. y e. B Pred ( R , B , y ) = (/) )

Distinct variable groups:   y, B   y, R

Allowed substitution hint:   A( y)

Proof of Theorem frmin

Dummy variables b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frss 5081	. . . 4 |- ( B C_ A -> ( R Fr A -> R Fr B ) )
2		sess2 5083	. . . 4 |- ( B C_ A -> ( R Se A -> R Se B ) )
3	1, 2	anim12d 586	. . 3 |- ( B C_ A -> ( ( R Fr A /\ R Se A ) -> ( R Fr B /\ R Se B ) ) )
4		n0 3931	. . . 4 |- ( B =/= (/) <-> E. b b e. B )
5		predeq3 5684	. . . . . . . . . . 11 |- ( y = b -> Pred ( R , B , y ) = Pred ( R , B , b ) )
6	5	eqeq1d 2624	. . . . . . . . . 10 |- ( y = b -> ( Pred ( R , B , y ) = (/) <-> Pred ( R , B , b ) = (/) ) )
7	6	rspcev 3309	. . . . . . . . 9 |- ( ( b e. B /\ Pred ( R , B , b ) = (/) ) -> E. y e. B Pred ( R , B , y ) = (/) )
8	7	ex 450	. . . . . . . 8 |- ( b e. B -> ( Pred ( R , B , b ) = (/) -> E. y e. B Pred ( R , B , y ) = (/) ) )
9	8	adantl 482	. . . . . . 7 |- ( ( ( R Fr B /\ R Se B ) /\ b e. B ) -> ( Pred ( R , B , b ) = (/) -> E. y e. B Pred ( R , B , y ) = (/) ) )
10		setlikespec 5701	. . . . . . . . . . 11 |- ( ( b e. B /\ R Se B ) -> Pred ( R , B , b ) e. _V )
11		trpredpred 31728	. . . . . . . . . . . . 13 |- ( Pred ( R , B , b ) e. _V -> Pred ( R , B , b ) C_ TrPred ( R , B , b ) )
12		ssn0 3976	. . . . . . . . . . . . . 14 |- ( ( Pred ( R , B , b ) C_ TrPred ( R , B , b ) /\ Pred ( R , B , b ) =/= (/) ) -> TrPred ( R , B , b ) =/= (/) )
13	12	ex 450	. . . . . . . . . . . . 13 |- ( Pred ( R , B , b ) C_ TrPred ( R , B , b ) -> ( Pred ( R , B , b ) =/= (/) -> TrPred ( R , B , b ) =/= (/) ) )
14	11, 13	syl 17	. . . . . . . . . . . 12 |- ( Pred ( R , B , b ) e. _V -> ( Pred ( R , B , b ) =/= (/) -> TrPred ( R , B , b ) =/= (/) ) )
15		trpredss 31729	. . . . . . . . . . . 12 |- ( Pred ( R , B , b ) e. _V -> TrPred ( R , B , b ) C_ B )
16	14, 15	jctild 566	. . . . . . . . . . 11 |- ( Pred ( R , B , b ) e. _V -> ( Pred ( R , B , b ) =/= (/) -> ( TrPred ( R , B , b ) C_ B /\ TrPred ( R , B , b ) =/= (/) ) ) )
17	10, 16	syl 17	. . . . . . . . . 10 |- ( ( b e. B /\ R Se B ) -> ( Pred ( R , B , b ) =/= (/) -> ( TrPred ( R , B , b ) C_ B /\ TrPred ( R , B , b ) =/= (/) ) ) )
18	17	adantr 481	. . . . . . . . 9 |- ( ( ( b e. B /\ R Se B ) /\ R Fr B ) -> ( Pred ( R , B , b ) =/= (/) -> ( TrPred ( R , B , b ) C_ B /\ TrPred ( R , B , b ) =/= (/) ) ) )
19		trpredex 31737	. . . . . . . . . . 11 |- TrPred ( R , B , b ) e. _V
20		sseq1 3626	. . . . . . . . . . . . . 14 |- ( c = TrPred ( R , B , b ) -> ( c C_ B <-> TrPred ( R , B , b ) C_ B ) )
21		neeq1 2856	. . . . . . . . . . . . . 14 |- ( c = TrPred ( R , B , b ) -> ( c =/= (/) <-> TrPred ( R , B , b ) =/= (/) ) )
22	20, 21	anbi12d 747	. . . . . . . . . . . . 13 |- ( c = TrPred ( R , B , b ) -> ( ( c C_ B /\ c =/= (/) ) <-> ( TrPred ( R , B , b ) C_ B /\ TrPred ( R , B , b ) =/= (/) ) ) )
23		predeq2 5683	. . . . . . . . . . . . . . 15 |- ( c = TrPred ( R , B , b ) -> Pred ( R , c , y ) = Pred ( R , TrPred ( R , B , b ) , y ) )
24	23	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( c = TrPred ( R , B , b ) -> ( Pred ( R , c , y ) = (/) <-> Pred ( R , TrPred ( R , B , b ) , y ) = (/) ) )
25	24	rexeqbi1dv 3147	. . . . . . . . . . . . 13 |- ( c = TrPred ( R , B , b ) -> ( E. y e. c Pred ( R , c , y ) = (/) <-> E. y e. TrPred ( R , B , b ) Pred ( R , TrPred ( R , B , b ) , y ) = (/) ) )
26	22, 25	imbi12d 334	. . . . . . . . . . . 12 |- ( c = TrPred ( R , B , b ) -> ( ( ( c C_ B /\ c =/= (/) ) -> E. y e. c Pred ( R , c , y ) = (/) ) <-> ( ( TrPred ( R , B , b ) C_ B /\ TrPred ( R , B , b ) =/= (/) ) -> E. y e. TrPred ( R , B , b ) Pred ( R , TrPred ( R , B , b ) , y ) = (/) ) ) )
27	26	imbi2d 330	. . . . . . . . . . 11 |- ( c = TrPred ( R , B , b ) -> ( ( R Fr B -> ( ( c C_ B /\ c =/= (/) ) -> E. y e. c Pred ( R , c , y ) = (/) ) ) <-> ( R Fr B -> ( ( TrPred ( R , B , b ) C_ B /\ TrPred ( R , B , b ) =/= (/) ) -> E. y e. TrPred ( R , B , b ) Pred ( R , TrPred ( R , B , b ) , y ) = (/) ) ) ) )
28		dffr4 5696	. . . . . . . . . . . 12 |- ( R Fr B <-> A. c ( ( c C_ B /\ c =/= (/) ) -> E. y e. c Pred ( R , c , y ) = (/) ) )
29		sp 2053	. . . . . . . . . . . 12 |- ( A. c ( ( c C_ B /\ c =/= (/) ) -> E. y e. c Pred ( R , c , y ) = (/) ) -> ( ( c C_ B /\ c =/= (/) ) -> E. y e. c Pred ( R , c , y ) = (/) ) )
30	28, 29	sylbi 207	. . . . . . . . . . 11 |- ( R Fr B -> ( ( c C_ B /\ c =/= (/) ) -> E. y e. c Pred ( R , c , y ) = (/) ) )
31	19, 27, 30	vtocl 3259	. . . . . . . . . 10 |- ( R Fr B -> ( ( TrPred ( R , B , b ) C_ B /\ TrPred ( R , B , b ) =/= (/) ) -> E. y e. TrPred ( R , B , b ) Pred ( R , TrPred ( R , B , b ) , y ) = (/) ) )
32	10, 15	syl 17	. . . . . . . . . . 11 |- ( ( b e. B /\ R Se B ) -> TrPred ( R , B , b ) C_ B )
33	32	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( b e. B /\ R Se B ) /\ y e. TrPred ( R , B , b ) ) -> TrPred ( R , B , b ) C_ B )
34		trpredtr 31730	. . . . . . . . . . . . . . . 16 |- ( ( b e. B /\ R Se B ) -> ( y e. TrPred ( R , B , b ) -> Pred ( R , B , y ) C_ TrPred ( R , B , b ) ) )
35	34	imp 445	. . . . . . . . . . . . . . 15 |- ( ( ( b e. B /\ R Se B ) /\ y e. TrPred ( R , B , b ) ) -> Pred ( R , B , y ) C_ TrPred ( R , B , b ) )
36		sspred 5688	. . . . . . . . . . . . . . 15 |- ( ( TrPred ( R , B , b ) C_ B /\ Pred ( R , B , y ) C_ TrPred ( R , B , b ) ) -> Pred ( R , B , y ) = Pred ( R , TrPred ( R , B , b ) , y ) )
37	33, 35, 36	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( b e. B /\ R Se B ) /\ y e. TrPred ( R , B , b ) ) -> Pred ( R , B , y ) = Pred ( R , TrPred ( R , B , b ) , y ) )
38	37	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( ( ( b e. B /\ R Se B ) /\ y e. TrPred ( R , B , b ) ) -> ( Pred ( R , B , y ) = (/) <-> Pred ( R , TrPred ( R , B , b ) , y ) = (/) ) )
39	38	biimprd 238	. . . . . . . . . . . 12 |- ( ( ( b e. B /\ R Se B ) /\ y e. TrPred ( R , B , b ) ) -> ( Pred ( R , TrPred ( R , B , b ) , y ) = (/) -> Pred ( R , B , y ) = (/) ) )
40	39	reximdva 3017	. . . . . . . . . . 11 |- ( ( b e. B /\ R Se B ) -> ( E. y e. TrPred ( R , B , b ) Pred ( R , TrPred ( R , B , b ) , y ) = (/) -> E. y e. TrPred ( R , B , b ) Pred ( R , B , y ) = (/) ) )
41		ssrexv 3667	. . . . . . . . . . 11 |- ( TrPred ( R , B , b ) C_ B -> ( E. y e. TrPred ( R , B , b ) Pred ( R , B , y ) = (/) -> E. y e. B Pred ( R , B , y ) = (/) ) )
42	32, 40, 41	sylsyld 61	. . . . . . . . . 10 |- ( ( b e. B /\ R Se B ) -> ( E. y e. TrPred ( R , B , b ) Pred ( R , TrPred ( R , B , b ) , y ) = (/) -> E. y e. B Pred ( R , B , y ) = (/) ) )
43	31, 42	sylan9r 690	. . . . . . . . 9 |- ( ( ( b e. B /\ R Se B ) /\ R Fr B ) -> ( ( TrPred ( R , B , b ) C_ B /\ TrPred ( R , B , b ) =/= (/) ) -> E. y e. B Pred ( R , B , y ) = (/) ) )
44	18, 43	syld 47	. . . . . . . 8 |- ( ( ( b e. B /\ R Se B ) /\ R Fr B ) -> ( Pred ( R , B , b ) =/= (/) -> E. y e. B Pred ( R , B , y ) = (/) ) )
45	44	an31s 848	. . . . . . 7 |- ( ( ( R Fr B /\ R Se B ) /\ b e. B ) -> ( Pred ( R , B , b ) =/= (/) -> E. y e. B Pred ( R , B , y ) = (/) ) )
46	9, 45	pm2.61dne 2880	. . . . . 6 |- ( ( ( R Fr B /\ R Se B ) /\ b e. B ) -> E. y e. B Pred ( R , B , y ) = (/) )
47	46	ex 450	. . . . 5 |- ( ( R Fr B /\ R Se B ) -> ( b e. B -> E. y e. B Pred ( R , B , y ) = (/) ) )
48	47	exlimdv 1861	. . . 4 |- ( ( R Fr B /\ R Se B ) -> ( E. b b e. B -> E. y e. B Pred ( R , B , y ) = (/) ) )
49	4, 48	syl5bi 232	. . 3 |- ( ( R Fr B /\ R Se B ) -> ( B =/= (/) -> E. y e. B Pred ( R , B , y ) = (/) ) )
50	3, 49	syl6com 37	. 2 |- ( ( R Fr A /\ R Se A ) -> ( B C_ A -> ( B =/= (/) -> E. y e. B Pred ( R , B , y ) = (/) ) ) )
51	50	imp32 449	1 |- ( ( ( R Fr A /\ R Se A ) /\ ( B C_ A /\ B =/= (/) ) ) -> E. y e. B Pred ( R , B , y ) = (/) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   Fr wfr 5070   Se wse 5071   Predcpred 5679   TrPredctrpred 31717

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

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-iun 4522  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-se 5074  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-trpred 31718

This theorem is referenced by:  frind  31740