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Theorem axpre-suploc 8217

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 8248. (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 529	. . 3 $/\$ $/\$ $/\$ $\/$
2		eleq1w 2293	. . . 4
3	2	cbvexv 1968	. . 3
4	1, 3	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$
5		simplll 535	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
6		simpr 110	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
7		simplrl 537	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$
8		breq2 4113	. . . . . . . 8
9	8	ralbidv 2542	. . . . . . 7
10	9	cbvrexv 2779	. . . . . 6
11		breq1 4112	. . . . . . . 8
12	11	cbvralv 2778	. . . . . . 7
13	12	rexbii 2549	. . . . . 6
14	10, 13	bitri 184	. . . . 5
15	7, 14	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
16		simplrr 538	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
17		breq1 4112	. . . . . . . 8
18		breq1 4112	. . . . . . . . . 10
19	18	rexbidv 2543	. . . . . . . . 9
20	19	orbi1d 799	. . . . . . . 8 $\/$ $\/$
21	17, 20	imbi12d 234	. . . . . . 7 $\/$ $\/$
22		breq2 4113	. . . . . . . 8
23		breq2 4113	. . . . . . . . . 10
24	23	ralbidv 2542	. . . . . . . . 9
25	24	orbi2d 798	. . . . . . . 8 $\/$ $\/$
26	22, 25	imbi12d 234	. . . . . . 7 $\/$ $\/$
27	21, 26	cbvral2v 2791	. . . . . 6 $\/$ $\/$
28		breq2 4113	. . . . . . . . . 10
29	28	cbvrexv 2779	. . . . . . . . 9
30		breq1 4112	. . . . . . . . . 10
31	30	cbvralv 2778	. . . . . . . . 9
32	29, 31	orbi12i 772	. . . . . . . 8 $\/$ $\/$
33	32	imbi2i 226	. . . . . . 7 $\/$ $\/$
34	33	2ralbii 2550	. . . . . 6 $\/$ $\/$
35	27, 34	bitri 184	. . . . 5 $\/$ $\/$
36	16, 35	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
37		eqid 2232	. . . 4
38	5, 6, 15, 36, 37	axpre-suploclemres 8216	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
39	17	notbid 673	. . . . . . . 8
40	39	ralbidv 2542	. . . . . . 7
41	8	imbi1d 231	. . . . . . . 8
42	41	ralbidv 2542	. . . . . . 7
43	40, 42	anbi12d 473	. . . . . 6 $/\$ $/\$
44	43	cbvrexv 2779	. . . . 5 $/\$ $/\$
45	22	notbid 673	. . . . . . . 8
46	45	cbvralv 2778	. . . . . . 7
47		breq1 4112	. . . . . . . . . 10
48	47	rexbidv 2543	. . . . . . . . 9
49	11, 48	imbi12d 234	. . . . . . . 8
50	49	cbvralv 2778	. . . . . . 7
51	46, 50	anbi12i 460	. . . . . 6 $/\$ $/\$
52	51	rexbii 2549	. . . . 5 $/\$ $/\$
53	44, 52	bitri 184	. . . 4 $/\$ $/\$
54		breq2 4113	. . . . . . . . 9
55	54	cbvrexv 2779	. . . . . . . 8
56	55	imbi2i 226	. . . . . . 7
57	56	ralbii 2548	. . . . . 6
58	57	anbi2i 457	. . . . 5 $/\$ $/\$
59	58	rexbii 2549	. . . 4 $/\$ $/\$
60	53, 59	bitri 184	. . 3 $/\$ $/\$
61	38, 60	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
62	4, 61	exlimddv 1948	1 $/\$ $/\$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 716

wex 1541

wcel 2203

wral 2520

wrex 2521

crab 2524

wss 3211

cop 3692 class class class wbr 4109

cnr 7612

c0r 7613

cr 8126

cltrr 8131

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-eprel 4410 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-irdg 6601 df-1o 6647 df-2o 6648 df-oadd 6651 df-omul 6652 df-er 6767 df-ec 6769 df-qs 6773 df-ni 7619 df-pli 7620 df-mi 7621 df-lti 7622 df-plpq 7659 df-mpq 7660 df-enq 7662 df-nqqs 7663 df-plqqs 7664 df-mqqs 7665 df-1nqqs 7666 df-rq 7667 df-ltnqqs 7668 df-enq0 7739 df-nq0 7740 df-0nq0 7741 df-plq0 7742 df-mq0 7743 df-inp 7781 df-i1p 7782 df-iplp 7783 df-imp 7784 df-iltp 7785 df-enr 8041 df-nr 8042 df-plr 8043 df-mr 8044 df-ltr 8045 df-0r 8046 df-1r 8047 df-m1r 8048 df-r 8137 df-lt 8140

This theorem is referenced by: (None)

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