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Theorem axpre-suploc 8165

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 8196. (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 2292	. . . 4
3	2	cbvexv 1967	. . 3
4	1, 3	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$
5		simplll 535	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
6		simpr 110	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
7		simplrl 537	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$
8		breq2 4097	. . . . . . . 8
9	8	ralbidv 2533	. . . . . . 7
10	9	cbvrexv 2769	. . . . . 6
11		breq1 4096	. . . . . . . 8
12	11	cbvralv 2768	. . . . . . 7
13	12	rexbii 2540	. . . . . 6
14	10, 13	bitri 184	. . . . 5
15	7, 14	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
16		simplrr 538	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
17		breq1 4096	. . . . . . . 8
18		breq1 4096	. . . . . . . . . 10
19	18	rexbidv 2534	. . . . . . . . 9
20	19	orbi1d 799	. . . . . . . 8 $\/$ $\/$
21	17, 20	imbi12d 234	. . . . . . 7 $\/$ $\/$
22		breq2 4097	. . . . . . . 8
23		breq2 4097	. . . . . . . . . 10
24	23	ralbidv 2533	. . . . . . . . 9
25	24	orbi2d 798	. . . . . . . 8 $\/$ $\/$
26	22, 25	imbi12d 234	. . . . . . 7 $\/$ $\/$
27	21, 26	cbvral2v 2781	. . . . . 6 $\/$ $\/$
28		breq2 4097	. . . . . . . . . 10
29	28	cbvrexv 2769	. . . . . . . . 9
30		breq1 4096	. . . . . . . . . 10
31	30	cbvralv 2768	. . . . . . . . 9
32	29, 31	orbi12i 772	. . . . . . . 8 $\/$ $\/$
33	32	imbi2i 226	. . . . . . 7 $\/$ $\/$
34	33	2ralbii 2541	. . . . . 6 $\/$ $\/$
35	27, 34	bitri 184	. . . . 5 $\/$ $\/$
36	16, 35	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
37		eqid 2231	. . . 4
38	5, 6, 15, 36, 37	axpre-suploclemres 8164	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
39	17	notbid 673	. . . . . . . 8
40	39	ralbidv 2533	. . . . . . 7
41	8	imbi1d 231	. . . . . . . 8
42	41	ralbidv 2533	. . . . . . 7
43	40, 42	anbi12d 473	. . . . . 6 $/\$ $/\$
44	43	cbvrexv 2769	. . . . 5 $/\$ $/\$
45	22	notbid 673	. . . . . . . 8
46	45	cbvralv 2768	. . . . . . 7
47		breq1 4096	. . . . . . . . . 10
48	47	rexbidv 2534	. . . . . . . . 9
49	11, 48	imbi12d 234	. . . . . . . 8
50	49	cbvralv 2768	. . . . . . 7
51	46, 50	anbi12i 460	. . . . . 6 $/\$ $/\$
52	51	rexbii 2540	. . . . 5 $/\$ $/\$
53	44, 52	bitri 184	. . . 4 $/\$ $/\$
54		breq2 4097	. . . . . . . . 9
55	54	cbvrexv 2769	. . . . . . . 8
56	55	imbi2i 226	. . . . . . 7
57	56	ralbii 2539	. . . . . 6
58	57	anbi2i 457	. . . . 5 $/\$ $/\$
59	58	rexbii 2540	. . . 4 $/\$ $/\$
60	53, 59	bitri 184	. . 3 $/\$ $/\$
61	38, 60	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
62	4, 61	exlimddv 1947	1 $/\$ $/\$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

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

wex 1541

wcel 2202

wral 2511

wrex 2512

crab 2515

wss 3201

cop 3676 class class class wbr 4093

cnr 7560

c0r 7561

cr 8074

cltrr 8079

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 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-eprel 4392 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-1o 6625 df-2o 6626 df-oadd 6629 df-omul 6630 df-er 6745 df-ec 6747 df-qs 6751 df-ni 7567 df-pli 7568 df-mi 7569 df-lti 7570 df-plpq 7607 df-mpq 7608 df-enq 7610 df-nqqs 7611 df-plqqs 7612 df-mqqs 7613 df-1nqqs 7614 df-rq 7615 df-ltnqqs 7616 df-enq0 7687 df-nq0 7688 df-0nq0 7689 df-plq0 7690 df-mq0 7691 df-inp 7729 df-i1p 7730 df-iplp 7731 df-imp 7732 df-iltp 7733 df-enr 7989 df-nr 7990 df-plr 7991 df-mr 7992 df-ltr 7993 df-0r 7994 df-1r 7995 df-m1r 7996 df-r 8085 df-lt 8088

This theorem is referenced by: (None)

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