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Theorem dedekindeulemlu 13393

Description: Lemma for dedekindeu 13395. 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 13392	. 2 $/\$
10		simpr 109	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
11	1	ad3antrrr 489	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
12	11, 10	sseldd 3148	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
13		rsp 2517	. . . . . . . . . . 11
14	5, 13	syl 14	. . . . . . . . . 10
15	14	ad3antrrr 489	. . . . . . . . 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 526	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21	19, 20	sseldd 3148	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		simp-4r 537	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		simprr 527	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		breq2 3993	. . . . . . . . . . 11
25	24	notbid 662	. . . . . . . . . 10
26		simprl 526	. . . . . . . . . . 11 $/\$ $/\$ $/\$
27	26	ad2antrr 485	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	25, 27, 20	rspcdva 2839	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	21, 22, 28	nltled 8040	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	18, 21, 22, 23, 29	ltletrd 8342	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31	17, 30	rexlimddv 2592	. . . . . 6 $/\$ $/\$ $/\$ $/\$
32	31	ralrimiva 2543	. . . . 5 $/\$ $/\$ $/\$
33		simpr 109	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
34		simplll 528	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	34, 2	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
36	35, 33	sseldd 3148	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
37		rsp 2517	. . . . . . . . . 10
38	6, 37	syl 14	. . . . . . . . 9
39	34, 36, 38	sylc 62	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
40	33, 39	mpbid 146	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41		simp-4r 537	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	35	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		simprl 526	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	42, 43	sseldd 3148	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	36	adantr 274	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	43	adantr 274	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47	34	ad2antrr 485	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48		simpr 109	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		breq1 3992	. . . . . . . . . . . . . . . 16
50		breq1 3992	. . . . . . . . . . . . . . . . 17
51	50	rexbidv 2471	. . . . . . . . . . . . . . . 16
52	49, 51	imbi12d 233	. . . . . . . . . . . . . . 15
53		simprr 527	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
54	53	ad3antrrr 489	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	44	adantr 274	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	52, 54, 55	rspcdva 2839	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	48, 56	mpd 13	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		breq2 3993	. . . . . . . . . . . . . 14
59	58	cbvrexv 2697	. . . . . . . . . . . . 13
60	57, 59	sylib 121	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61	47, 55, 14	sylc 62	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	60, 61	mpbird 166	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		disj 3463	. . . . . . . . . . . . 13
64	7, 63	sylib 121	. . . . . . . . . . . 12
65	64	r19.21bi 2558	. . . . . . . . . . 11 $/\$
66	47, 62, 65	syl2anc 409	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	46, 66	pm2.65da 656	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
68	41, 44, 67	nltled 8040	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69		simprr 527	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	41, 44, 45, 68, 69	lelttrd 8044	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	40, 70	rexlimddv 2592	. . . . . 6 $/\$ $/\$ $/\$ $/\$
72	71	ralrimiva 2543	. . . . 5 $/\$ $/\$ $/\$
73	32, 72	jca 304	. . . 4 $/\$ $/\$ $/\$ $/\$
74	73	ex 114	. . 3 $/\$ $/\$ $/\$
75	74	reximdva 2572	. 2 $/\$ $/\$
76	9, 75	mpd 13	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703

wceq 1348

wcel 2141

wral 2448

wrex 2449

cin 3120

wss 3121

c0 3414 class class class wbr 3989

cr 7773

clt 7954

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-pre-ltwlin 7887 ax-pre-suploc 7895

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-cnv 4619 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960

This theorem is referenced by: dedekindeu 13395

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