dedekindeulemlu - Intuitionistic Logic Explorer

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Theorem dedekindeulemlu 12807

Description: Lemma for dedekindeu 12809. 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 12806	. 2 $/\$
10		simpr 109	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
11	1	ad3antrrr 484	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
12	11, 10	sseldd 3103	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
13		rsp 2483	. . . . . . . . . . 11
14	5, 13	syl 14	. . . . . . . . . 10
15	14	ad3antrrr 484	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
16	12, 15	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
17	10, 16	mpbid 146	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
18	12	adantr 274	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	11	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		simprl 521	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21	19, 20	sseldd 3103	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		simp-4r 532	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		simprr 522	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		breq2 3941	. . . . . . . . . . 11
25	24	notbid 657	. . . . . . . . . 10
26		simprl 521	. . . . . . . . . . 11 $/\$ $/\$ $/\$
27	26	ad2antrr 480	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	25, 27, 20	rspcdva 2798	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	21, 22, 28	nltled 7907	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	18, 21, 22, 23, 29	ltletrd 8209	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31	17, 30	rexlimddv 2557	. . . . . 6 $/\$ $/\$ $/\$ $/\$
32	31	ralrimiva 2508	. . . . 5 $/\$ $/\$ $/\$
33		simpr 109	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
34		simplll 523	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	34, 2	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
36	35, 33	sseldd 3103	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
37		rsp 2483	. . . . . . . . . 10
38	6, 37	syl 14	. . . . . . . . 9
39	34, 36, 38	sylc 62	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
40	33, 39	mpbid 146	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41		simp-4r 532	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	35	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		simprl 521	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	42, 43	sseldd 3103	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	36	adantr 274	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	43	adantr 274	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47	34	ad2antrr 480	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48		simpr 109	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		breq1 3940	. . . . . . . . . . . . . . . 16
50		breq1 3940	. . . . . . . . . . . . . . . . 17
51	50	rexbidv 2439	. . . . . . . . . . . . . . . 16
52	49, 51	imbi12d 233	. . . . . . . . . . . . . . 15
53		simprr 522	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
54	53	ad3antrrr 484	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	44	adantr 274	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	52, 54, 55	rspcdva 2798	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	48, 56	mpd 13	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		breq2 3941	. . . . . . . . . . . . . 14
59	58	cbvrexv 2658	. . . . . . . . . . . . 13
60	57, 59	sylib 121	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61	47, 55, 14	sylc 62	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	60, 61	mpbird 166	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		disj 3416	. . . . . . . . . . . . 13
64	7, 63	sylib 121	. . . . . . . . . . . 12
65	64	r19.21bi 2523	. . . . . . . . . . 11 $/\$
66	47, 62, 65	syl2anc 409	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	46, 66	pm2.65da 651	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
68	41, 44, 67	nltled 7907	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69		simprr 522	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	41, 44, 45, 68, 69	lelttrd 7911	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	40, 70	rexlimddv 2557	. . . . . 6 $/\$ $/\$ $/\$ $/\$
72	71	ralrimiva 2508	. . . . 5 $/\$ $/\$ $/\$
73	32, 72	jca 304	. . . 4 $/\$ $/\$ $/\$ $/\$
74	73	ex 114	. . 3 $/\$ $/\$ $/\$
75	74	reximdva 2537	. 2 $/\$ $/\$
76	9, 75	mpd 13	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698

wceq 1332

wcel 1481

wral 2417

wrex 2418

cin 3075

wss 3076

c0 3368 class class class wbr 3937

cr 7643

clt 7824

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltwlin 7757 ax-pre-suploc 7765

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830

This theorem is referenced by: dedekindeu 12809

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