dedekindeulemlu - Intuitionistic Logic Explorer

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Theorem dedekindeulemlu 14857

Description: Lemma for dedekindeu 14859. 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 14856	. 2 $/\$
10		simpr 110	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
11	1	ad3antrrr 492	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
12	11, 10	sseldd 3184	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
13		rsp 2544	. . . . . . . . . . 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 3184	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		simp-4r 542	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		simprr 531	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		breq2 4037	. . . . . . . . . . 11
25	24	notbid 668	. . . . . . . . . 10
26		simprl 529	. . . . . . . . . . 11 $/\$ $/\$ $/\$
27	26	ad2antrr 488	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	25, 27, 20	rspcdva 2873	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	21, 22, 28	nltled 8147	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	18, 21, 22, 23, 29	ltletrd 8450	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31	17, 30	rexlimddv 2619	. . . . . 6 $/\$ $/\$ $/\$ $/\$
32	31	ralrimiva 2570	. . . . 5 $/\$ $/\$ $/\$
33		simpr 110	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
34		simplll 533	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	34, 2	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
36	35, 33	sseldd 3184	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
37		rsp 2544	. . . . . . . . . 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 3184	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	36	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	43	adantr 276	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47	34	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48		simpr 110	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		breq1 4036	. . . . . . . . . . . . . . . 16
50		breq1 4036	. . . . . . . . . . . . . . . . 17
51	50	rexbidv 2498	. . . . . . . . . . . . . . . 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 2873	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	48, 56	mpd 13	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		breq2 4037	. . . . . . . . . . . . . 14
59	58	cbvrexv 2730	. . . . . . . . . . . . 13
60	57, 59	sylib 122	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61	47, 55, 14	sylc 62	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	60, 61	mpbird 167	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		disj 3499	. . . . . . . . . . . . 13
64	7, 63	sylib 122	. . . . . . . . . . . 12
65	64	r19.21bi 2585	. . . . . . . . . . 11 $/\$
66	47, 62, 65	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	46, 66	pm2.65da 662	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
68	41, 44, 67	nltled 8147	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69		simprr 531	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	41, 44, 45, 68, 69	lelttrd 8151	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	40, 70	rexlimddv 2619	. . . . . 6 $/\$ $/\$ $/\$ $/\$
72	71	ralrimiva 2570	. . . . 5 $/\$ $/\$ $/\$
73	32, 72	jca 306	. . . 4 $/\$ $/\$ $/\$ $/\$
74	73	ex 115	. . 3 $/\$ $/\$ $/\$
75	74	reximdva 2599	. 2 $/\$ $/\$
76	9, 75	mpd 13	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709

wceq 1364

wcel 2167

wral 2475

wrex 2476

cin 3156

wss 3157

c0 3450 class class class wbr 4033

cr 7878

clt 8061

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971 ax-pre-ltwlin 7992 ax-pre-suploc 8000

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-rab 2484 df-v 2765 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-opab 4095 df-xp 4669 df-cnv 4671 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067

This theorem is referenced by: dedekindeu 14859

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