Theorem clsk1indlem1 38343

Description: The ansatz closure function ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) ) does not have the K1 property of isotony. (Contributed by RP, 6-Jul-2021.)

Hypothesis

Ref	Expression
clsk1indlem.k	|- K = ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) )

Assertion

Ref	Expression
clsk1indlem1	|- E. s e. ~P 3o E. t e. ~P 3o ( s C_ t /\ -. ( K ` s ) C_ ( K ` t ) )

Distinct variable groups:   K, s, t   s, r, t

Allowed substitution hint:   K( r)

Proof of Theorem clsk1indlem1

Step	Hyp	Ref	Expression
1		tpex 6957	. . . . . 6 |- { (/) , 1o , 2o } e. _V
2	1	a1i 11	. . . . 5 |- ( T. -> { (/) , 1o , 2o } e. _V )
3		snsstp1 4347	. . . . . 6 |- { (/) } C_ { (/) , 1o , 2o }
4	3	a1i 11	. . . . 5 |- ( T. -> { (/) } C_ { (/) , 1o , 2o } )
5	2, 4	sselpwd 4807	. . . 4 |- ( T. -> { (/) } e. ~P { (/) , 1o , 2o } )
6	5	trud 1493	. . 3 |- { (/) } e. ~P { (/) , 1o , 2o }
7		df3o2 38322	. . . 4 |- 3o = { (/) , 1o , 2o }
8	7	pweqi 4162	. . 3 |- ~P 3o = ~P { (/) , 1o , 2o }
9	6, 8	eleqtrri 2700	. 2 |- { (/) } e. ~P 3o
10		0ex 4790	. . . . . . . 8 |- (/) e. _V
11	10	snss 4316	. . . . . . 7 |- ( (/) e. { (/) , 1o , 2o } <-> { (/) } C_ { (/) , 1o , 2o } )
12	4, 11	sylibr 224	. . . . . 6 |- ( T. -> (/) e. { (/) , 1o , 2o } )
13		snsstp3 4349	. . . . . . . 8 |- { 2o } C_ { (/) , 1o , 2o }
14	13	a1i 11	. . . . . . 7 |- ( T. -> { 2o } C_ { (/) , 1o , 2o } )
15		2on 7568	. . . . . . . . 9 |- 2o e. On
16	15	elexi 3213	. . . . . . . 8 |- 2o e. _V
17	16	snss 4316	. . . . . . 7 |- ( 2o e. { (/) , 1o , 2o } <-> { 2o } C_ { (/) , 1o , 2o } )
18	14, 17	sylibr 224	. . . . . 6 |- ( T. -> 2o e. { (/) , 1o , 2o } )
19	12, 18	prssd 4354	. . . . 5 |- ( T. -> { (/) , 2o } C_ { (/) , 1o , 2o } )
20	2, 19	sselpwd 4807	. . . 4 |- ( T. -> { (/) , 2o } e. ~P { (/) , 1o , 2o } )
21	20	trud 1493	. . 3 |- { (/) , 2o } e. ~P { (/) , 1o , 2o }
22	21, 8	eleqtrri 2700	. 2 |- { (/) , 2o } e. ~P 3o
23		simpl 473	. . 3 |- ( ( { (/) } e. ~P 3o /\ { (/) , 2o } e. ~P 3o ) -> { (/) } e. ~P 3o )
24		sseq1 3626	. . . . . 6 |- ( s = { (/) } -> ( s C_ t <-> { (/) } C_ t ) )
25		fveq2 6191	. . . . . . . 8 |- ( s = { (/) } -> ( K ` s ) = ( K ` { (/) } ) )
26	25	sseq1d 3632	. . . . . . 7 |- ( s = { (/) } -> ( ( K ` s ) C_ ( K ` t ) <-> ( K ` { (/) } ) C_ ( K ` t ) ) )
27	26	notbid 308	. . . . . 6 |- ( s = { (/) } -> ( -. ( K ` s ) C_ ( K ` t ) <-> -. ( K ` { (/) } ) C_ ( K ` t ) ) )
28	24, 27	anbi12d 747	. . . . 5 |- ( s = { (/) } -> ( ( s C_ t /\ -. ( K ` s ) C_ ( K ` t ) ) <-> ( { (/) } C_ t /\ -. ( K ` { (/) } ) C_ ( K ` t ) ) ) )
29	28	rexbidv 3052	. . . 4 |- ( s = { (/) } -> ( E. t e. ~P 3o ( s C_ t /\ -. ( K ` s ) C_ ( K ` t ) ) <-> E. t e. ~P 3o ( { (/) } C_ t /\ -. ( K ` { (/) } ) C_ ( K ` t ) ) ) )
30	29	adantl 482	. . 3 |- ( ( ( { (/) } e. ~P 3o /\ { (/) , 2o } e. ~P 3o ) /\ s = { (/) } ) -> ( E. t e. ~P 3o ( s C_ t /\ -. ( K ` s ) C_ ( K ` t ) ) <-> E. t e. ~P 3o ( { (/) } C_ t /\ -. ( K ` { (/) } ) C_ ( K ` t ) ) ) )
31		simpr 477	. . . 4 |- ( ( { (/) } e. ~P 3o /\ { (/) , 2o } e. ~P 3o ) -> { (/) , 2o } e. ~P 3o )
32		fveq2 6191	. . . . . . . 8 |- ( t = { (/) , 2o } -> ( K ` t ) = ( K ` { (/) , 2o } ) )
33	32	sseq2d 3633	. . . . . . 7 |- ( t = { (/) , 2o } -> ( ( K ` { (/) } ) C_ ( K ` t ) <-> ( K ` { (/) } ) C_ ( K ` { (/) , 2o } ) ) )
34	33	notbid 308	. . . . . 6 |- ( t = { (/) , 2o } -> ( -. ( K ` { (/) } ) C_ ( K ` t ) <-> -. ( K ` { (/) } ) C_ ( K ` { (/) , 2o } ) ) )
35	34	cleq2lem 37914	. . . . 5 |- ( t = { (/) , 2o } -> ( ( { (/) } C_ t /\ -. ( K ` { (/) } ) C_ ( K ` t ) ) <-> ( { (/) } C_ { (/) , 2o } /\ -. ( K ` { (/) } ) C_ ( K ` { (/) , 2o } ) ) ) )
36	35	adantl 482	. . . 4 |- ( ( ( { (/) } e. ~P 3o /\ { (/) , 2o } e. ~P 3o ) /\ t = { (/) , 2o } ) -> ( ( { (/) } C_ t /\ -. ( K ` { (/) } ) C_ ( K ` t ) ) <-> ( { (/) } C_ { (/) , 2o } /\ -. ( K ` { (/) } ) C_ ( K ` { (/) , 2o } ) ) ) )
37		1on 7567	. . . . . . . . 9 |- 1o e. On
38	37	elexi 3213	. . . . . . . 8 |- 1o e. _V
39	38	prid2 4298	. . . . . . 7 |- 1o e. { (/) , 1o }
40		iftrue 4092	. . . . . . . . 9 |- ( r = { (/) } -> if ( r = { (/) } , { (/) , 1o } , r ) = { (/) , 1o } )
41		clsk1indlem.k	. . . . . . . . 9 |- K = ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) )
42		prex 4909	. . . . . . . . 9 |- { (/) , 1o } e. _V
43	40, 41, 42	fvmpt 6282	. . . . . . . 8 |- ( { (/) } e. ~P 3o -> ( K ` { (/) } ) = { (/) , 1o } )
44	43	adantr 481	. . . . . . 7 |- ( ( { (/) } e. ~P 3o /\ { (/) , 2o } e. ~P 3o ) -> ( K ` { (/) } ) = { (/) , 1o } )
45	39, 44	syl5eleqr 2708	. . . . . 6 |- ( ( { (/) } e. ~P 3o /\ { (/) , 2o } e. ~P 3o ) -> 1o e. ( K ` { (/) } ) )
46		1n0 7575	. . . . . . . . . . 11 |- 1o =/= (/)
47	46	neii 2796	. . . . . . . . . 10 |- -. 1o = (/)
48		eqcom 2629	. . . . . . . . . . . 12 |- ( 1o = 2o <-> 2o = 1o )
49		df-2o 7561	. . . . . . . . . . . . 13 |- 2o = suc 1o
50		df-1o 7560	. . . . . . . . . . . . 13 |- 1o = suc (/)
51	49, 50	eqeq12i 2636	. . . . . . . . . . . 12 |- ( 2o = 1o <-> suc 1o = suc (/) )
52		suc11reg 8516	. . . . . . . . . . . 12 |- ( suc 1o = suc (/) <-> 1o = (/) )
53	48, 51, 52	3bitri 286	. . . . . . . . . . 11 |- ( 1o = 2o <-> 1o = (/) )
54	46, 53	nemtbir 2889	. . . . . . . . . 10 |- -. 1o = 2o
55	47, 54	pm3.2ni 899	. . . . . . . . 9 |- -. ( 1o = (/) \/ 1o = 2o )
56		elpri 4197	. . . . . . . . 9 |- ( 1o e. { (/) , 2o } -> ( 1o = (/) \/ 1o = 2o ) )
57	55, 56	mto 188	. . . . . . . 8 |- -. 1o e. { (/) , 2o }
58	57	a1i 11	. . . . . . 7 |- ( ( { (/) } e. ~P 3o /\ { (/) , 2o } e. ~P 3o ) -> -. 1o e. { (/) , 2o } )
59		eqeq1 2626	. . . . . . . . . . 11 |- ( r = { (/) , 2o } -> ( r = { (/) } <-> { (/) , 2o } = { (/) } ) )
60		id 22	. . . . . . . . . . 11 |- ( r = { (/) , 2o } -> r = { (/) , 2o } )
61	59, 60	ifbieq2d 4111	. . . . . . . . . 10 |- ( r = { (/) , 2o } -> if ( r = { (/) } , { (/) , 1o } , r ) = if ( { (/) , 2o } = { (/) } , { (/) , 1o } , { (/) , 2o } ) )
62	16	prid2 4298	. . . . . . . . . . . 12 |- 2o e. { (/) , 2o }
63		2on0 7569	. . . . . . . . . . . . 13 |- 2o =/= (/)
64		nelsn 4212	. . . . . . . . . . . . 13 |- ( 2o =/= (/) -> -. 2o e. { (/) } )
65	63, 64	ax-mp 5	. . . . . . . . . . . 12 |- -. 2o e. { (/) }
66		nelneq2 2726	. . . . . . . . . . . 12 |- ( ( 2o e. { (/) , 2o } /\ -. 2o e. { (/) } ) -> -. { (/) , 2o } = { (/) } )
67	62, 65, 66	mp2an 708	. . . . . . . . . . 11 |- -. { (/) , 2o } = { (/) }
68	67	iffalsei 4096	. . . . . . . . . 10 |- if ( { (/) , 2o } = { (/) } , { (/) , 1o } , { (/) , 2o } ) = { (/) , 2o }
69	61, 68	syl6eq 2672	. . . . . . . . 9 |- ( r = { (/) , 2o } -> if ( r = { (/) } , { (/) , 1o } , r ) = { (/) , 2o } )
70		prex 4909	. . . . . . . . 9 |- { (/) , 2o } e. _V
71	69, 41, 70	fvmpt 6282	. . . . . . . 8 |- ( { (/) , 2o } e. ~P 3o -> ( K ` { (/) , 2o } ) = { (/) , 2o } )
72	71	adantl 482	. . . . . . 7 |- ( ( { (/) } e. ~P 3o /\ { (/) , 2o } e. ~P 3o ) -> ( K ` { (/) , 2o } ) = { (/) , 2o } )
73	58, 72	neleqtrrd 2723	. . . . . 6 |- ( ( { (/) } e. ~P 3o /\ { (/) , 2o } e. ~P 3o ) -> -. 1o e. ( K ` { (/) , 2o } ) )
74		nelss 3664	. . . . . 6 |- ( ( 1o e. ( K ` { (/) } ) /\ -. 1o e. ( K ` { (/) , 2o } ) ) -> -. ( K ` { (/) } ) C_ ( K ` { (/) , 2o } ) )
75	45, 73, 74	syl2anc 693	. . . . 5 |- ( ( { (/) } e. ~P 3o /\ { (/) , 2o } e. ~P 3o ) -> -. ( K ` { (/) } ) C_ ( K ` { (/) , 2o } ) )
76		snsspr1 4345	. . . . 5 |- { (/) } C_ { (/) , 2o }
77	75, 76	jctil 560	. . . 4 |- ( ( { (/) } e. ~P 3o /\ { (/) , 2o } e. ~P 3o ) -> ( { (/) } C_ { (/) , 2o } /\ -. ( K ` { (/) } ) C_ ( K ` { (/) , 2o } ) ) )
78	31, 36, 77	rspcedvd 3317	. . 3 |- ( ( { (/) } e. ~P 3o /\ { (/) , 2o } e. ~P 3o ) -> E. t e. ~P 3o ( { (/) } C_ t /\ -. ( K ` { (/) } ) C_ ( K ` t ) ) )
79	23, 30, 78	rspcedvd 3317	. 2 |- ( ( { (/) } e. ~P 3o /\ { (/) , 2o } e. ~P 3o ) -> E. s e. ~P 3o E. t e. ~P 3o ( s C_ t /\ -. ( K ` s ) C_ ( K ` t ) ) )
80	9, 22, 79	mp2an 708	1 |- E. s e. ~P 3o E. t e. ~P 3o ( s C_ t /\ -. ( K ` s ) C_ ( K ` t ) )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   {cpr 4179   {ctp 4181   |-> cmpt 4729   Oncon0 5723   suc csuc 5725   ` cfv 5888   1oc1o 7553   2oc2o 7554   3oc3o 7555

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-pr 4906  ax-un 6949  ax-reg 8497

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-rab 2921  df-v 3202  df-sbc 3436  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-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-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896  df-1o 7560  df-2o 7561  df-3o 7562

This theorem is referenced by:  clsk1independent  38344