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Theorem axpre-suploc 7710

Description: An inhabited, bounded-above, located set of reals has a supremum.

Locatedness here means that given , either there is an element of the set greater than , or is an upper bound.

This construction-dependent theorem should not be referenced directly; instead, use ax-pre-suploc 7741. (Contributed by Jim Kingdon, 23-Jan-2024.) (New usage is discouraged.)

**Assertion**
Ref	Expression
axpre-suploc	$/\$ $/\$ $/\$ $\/$ $/\$

Distinct variable group:

Proof of Theorem axpre-suploc Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 519	. . 3 $/\$ $/\$ $/\$ $\/$
2		eleq1w 2200	. . . 4
3	2	cbvexv 1890	. . 3
4	1, 3	sylib 121	. 2 $/\$ $/\$ $/\$ $\/$
5		simplll 522	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
6		simpr 109	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
7		simplrl 524	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$
8		breq2 3933	. . . . . . . 8
9	8	ralbidv 2437	. . . . . . 7
10	9	cbvrexv 2655	. . . . . 6
11		breq1 3932	. . . . . . . 8
12	11	cbvralv 2654	. . . . . . 7
13	12	rexbii 2442	. . . . . 6
14	10, 13	bitri 183	. . . . 5
15	7, 14	sylibr 133	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
16		simplrr 525	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
17		breq1 3932	. . . . . . . 8
18		breq1 3932	. . . . . . . . . 10
19	18	rexbidv 2438	. . . . . . . . 9
20	19	orbi1d 780	. . . . . . . 8 $\/$ $\/$
21	17, 20	imbi12d 233	. . . . . . 7 $\/$ $\/$
22		breq2 3933	. . . . . . . 8
23		breq2 3933	. . . . . . . . . 10
24	23	ralbidv 2437	. . . . . . . . 9
25	24	orbi2d 779	. . . . . . . 8 $\/$ $\/$
26	22, 25	imbi12d 233	. . . . . . 7 $\/$ $\/$
27	21, 26	cbvral2v 2665	. . . . . 6 $\/$ $\/$
28		breq2 3933	. . . . . . . . . 10
29	28	cbvrexv 2655	. . . . . . . . 9
30		breq1 3932	. . . . . . . . . 10
31	30	cbvralv 2654	. . . . . . . . 9
32	29, 31	orbi12i 753	. . . . . . . 8 $\/$ $\/$
33	32	imbi2i 225	. . . . . . 7 $\/$ $\/$
34	33	2ralbii 2443	. . . . . 6 $\/$ $\/$
35	27, 34	bitri 183	. . . . 5 $\/$ $\/$
36	16, 35	sylibr 133	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
37		eqid 2139	. . . 4
38	5, 6, 15, 36, 37	axpre-suploclemres 7709	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
39	17	notbid 656	. . . . . . . 8
40	39	ralbidv 2437	. . . . . . 7
41	8	imbi1d 230	. . . . . . . 8
42	41	ralbidv 2437	. . . . . . 7
43	40, 42	anbi12d 464	. . . . . 6 $/\$ $/\$
44	43	cbvrexv 2655	. . . . 5 $/\$ $/\$
45	22	notbid 656	. . . . . . . 8
46	45	cbvralv 2654	. . . . . . 7
47		breq1 3932	. . . . . . . . . 10
48	47	rexbidv 2438	. . . . . . . . 9
49	11, 48	imbi12d 233	. . . . . . . 8
50	49	cbvralv 2654	. . . . . . 7
51	46, 50	anbi12i 455	. . . . . 6 $/\$ $/\$
52	51	rexbii 2442	. . . . 5 $/\$ $/\$
53	44, 52	bitri 183	. . . 4 $/\$ $/\$
54		breq2 3933	. . . . . . . . 9
55	54	cbvrexv 2655	. . . . . . . 8
56	55	imbi2i 225	. . . . . . 7
57	56	ralbii 2441	. . . . . 6
58	57	anbi2i 452	. . . . 5 $/\$ $/\$
59	58	rexbii 2442	. . . 4 $/\$ $/\$
60	53, 59	bitri 183	. . 3 $/\$ $/\$
61	38, 60	sylib 121	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
62	4, 61	exlimddv 1870	1 $/\$ $/\$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103 $\/$ wo 697

wex 1468

wcel 1480

wral 2416

wrex 2417

crab 2420

wss 3071

cop 3530 class class class wbr 3929

cnr 7105

c0r 7106

cr 7619

cltrr 7624

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-2o 6314 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-enq0 7232 df-nq0 7233 df-0nq0 7234 df-plq0 7235 df-mq0 7236 df-inp 7274 df-i1p 7275 df-iplp 7276 df-imp 7277 df-iltp 7278 df-enr 7534 df-nr 7535 df-plr 7536 df-mr 7537 df-ltr 7538 df-0r 7539 df-1r 7540 df-m1r 7541 df-r 7630 df-lt 7633

This theorem is referenced by: (None)

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