dedekindeulemlu - Intuitionistic Logic Explorer

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Theorem dedekindeulemlu 13766

Description: Lemma for dedekindeu 13768. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 31-Jan-2024.)

**Hypotheses**
Ref	Expression
dedekindeu.lss
dedekindeu.uss
dedekindeu.lm
dedekindeu.um
dedekindeu.lr
dedekindeu.ur
dedekindeu.disj
dedekindeu.loc	$\/$

**Assertion**
Ref	Expression
dedekindeulemlu	$/\$

Distinct variable groups:

Allowed substitution hint:

(

)

Proof of Theorem dedekindeulemlu Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dedekindeu.lss	. . 3
2		dedekindeu.uss	. . 3
3		dedekindeu.lm	. . 3
4		dedekindeu.um	. . 3
5		dedekindeu.lr	. . 3
6		dedekindeu.ur	. . 3
7		dedekindeu.disj	. . 3
8		dedekindeu.loc	. . 3 $\/$
9	1, 2, 3, 4, 5, 6, 7, 8	dedekindeulemlub 13765	. 2 $/\$
10		simpr 110	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
11	1	ad3antrrr 492	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
12	11, 10	sseldd 3156	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
13		rsp 2524	. . . . . . . . . . 11
14	5, 13	syl 14	. . . . . . . . . 10
15	14	ad3antrrr 492	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
16	12, 15	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
17	10, 16	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
18	12	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	11	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21	19, 20	sseldd 3156	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		simp-4r 542	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		simprr 531	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		breq2 4004	. . . . . . . . . . 11
25	24	notbid 667	. . . . . . . . . 10
26		simprl 529	. . . . . . . . . . 11 $/\$ $/\$ $/\$
27	26	ad2antrr 488	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	25, 27, 20	rspcdva 2846	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	21, 22, 28	nltled 8068	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	18, 21, 22, 23, 29	ltletrd 8370	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31	17, 30	rexlimddv 2599	. . . . . 6 $/\$ $/\$ $/\$ $/\$
32	31	ralrimiva 2550	. . . . 5 $/\$ $/\$ $/\$
33		simpr 110	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
34		simplll 533	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	34, 2	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
36	35, 33	sseldd 3156	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
37		rsp 2524	. . . . . . . . . 10
38	6, 37	syl 14	. . . . . . . . 9
39	34, 36, 38	sylc 62	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
40	33, 39	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41		simp-4r 542	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	35	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	42, 43	sseldd 3156	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	36	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	43	adantr 276	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47	34	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48		simpr 110	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		breq1 4003	. . . . . . . . . . . . . . . 16
50		breq1 4003	. . . . . . . . . . . . . . . . 17
51	50	rexbidv 2478	. . . . . . . . . . . . . . . 16
52	49, 51	imbi12d 234	. . . . . . . . . . . . . . 15
53		simprr 531	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
54	53	ad3antrrr 492	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	44	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	52, 54, 55	rspcdva 2846	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	48, 56	mpd 13	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		breq2 4004	. . . . . . . . . . . . . 14
59	58	cbvrexv 2704	. . . . . . . . . . . . 13
60	57, 59	sylib 122	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61	47, 55, 14	sylc 62	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	60, 61	mpbird 167	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		disj 3471	. . . . . . . . . . . . 13
64	7, 63	sylib 122	. . . . . . . . . . . 12
65	64	r19.21bi 2565	. . . . . . . . . . 11 $/\$
66	47, 62, 65	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	46, 66	pm2.65da 661	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
68	41, 44, 67	nltled 8068	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69		simprr 531	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	41, 44, 45, 68, 69	lelttrd 8072	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	40, 70	rexlimddv 2599	. . . . . 6 $/\$ $/\$ $/\$ $/\$
72	71	ralrimiva 2550	. . . . 5 $/\$ $/\$ $/\$
73	32, 72	jca 306	. . . 4 $/\$ $/\$ $/\$ $/\$
74	73	ex 115	. . 3 $/\$ $/\$ $/\$
75	74	reximdva 2579	. 2 $/\$ $/\$
76	9, 75	mpd 13	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 708

wceq 1353

wcel 2148

wral 2455

wrex 2456

cin 3128

wss 3129

c0 3422 class class class wbr 4000

cr 7801

clt 7982

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-pre-ltwlin 7915 ax-pre-suploc 7923

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-xp 4629 df-cnv 4631 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988

This theorem is referenced by: dedekindeu 13768

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