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Theorem axpre-suploc 8045

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 8076. (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 528	. . 3 $/\$ $/\$ $/\$ $\/$
2		eleq1w 2267	. . . 4
3	2	cbvexv 1943	. . 3
4	1, 3	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$
5		simplll 533	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
6		simpr 110	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
7		simplrl 535	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$
8		breq2 4058	. . . . . . . 8
9	8	ralbidv 2507	. . . . . . 7
10	9	cbvrexv 2740	. . . . . 6
11		breq1 4057	. . . . . . . 8
12	11	cbvralv 2739	. . . . . . 7
13	12	rexbii 2514	. . . . . 6
14	10, 13	bitri 184	. . . . 5
15	7, 14	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
16		simplrr 536	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
17		breq1 4057	. . . . . . . 8
18		breq1 4057	. . . . . . . . . 10
19	18	rexbidv 2508	. . . . . . . . 9
20	19	orbi1d 793	. . . . . . . 8 $\/$ $\/$
21	17, 20	imbi12d 234	. . . . . . 7 $\/$ $\/$
22		breq2 4058	. . . . . . . 8
23		breq2 4058	. . . . . . . . . 10
24	23	ralbidv 2507	. . . . . . . . 9
25	24	orbi2d 792	. . . . . . . 8 $\/$ $\/$
26	22, 25	imbi12d 234	. . . . . . 7 $\/$ $\/$
27	21, 26	cbvral2v 2752	. . . . . 6 $\/$ $\/$
28		breq2 4058	. . . . . . . . . 10
29	28	cbvrexv 2740	. . . . . . . . 9
30		breq1 4057	. . . . . . . . . 10
31	30	cbvralv 2739	. . . . . . . . 9
32	29, 31	orbi12i 766	. . . . . . . 8 $\/$ $\/$
33	32	imbi2i 226	. . . . . . 7 $\/$ $\/$
34	33	2ralbii 2515	. . . . . 6 $\/$ $\/$
35	27, 34	bitri 184	. . . . 5 $\/$ $\/$
36	16, 35	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
37		eqid 2206	. . . 4
38	5, 6, 15, 36, 37	axpre-suploclemres 8044	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
39	17	notbid 669	. . . . . . . 8
40	39	ralbidv 2507	. . . . . . 7
41	8	imbi1d 231	. . . . . . . 8
42	41	ralbidv 2507	. . . . . . 7
43	40, 42	anbi12d 473	. . . . . 6 $/\$ $/\$
44	43	cbvrexv 2740	. . . . 5 $/\$ $/\$
45	22	notbid 669	. . . . . . . 8
46	45	cbvralv 2739	. . . . . . 7
47		breq1 4057	. . . . . . . . . 10
48	47	rexbidv 2508	. . . . . . . . 9
49	11, 48	imbi12d 234	. . . . . . . 8
50	49	cbvralv 2739	. . . . . . 7
51	46, 50	anbi12i 460	. . . . . 6 $/\$ $/\$
52	51	rexbii 2514	. . . . 5 $/\$ $/\$
53	44, 52	bitri 184	. . . 4 $/\$ $/\$
54		breq2 4058	. . . . . . . . 9
55	54	cbvrexv 2740	. . . . . . . 8
56	55	imbi2i 226	. . . . . . 7
57	56	ralbii 2513	. . . . . 6
58	57	anbi2i 457	. . . . 5 $/\$ $/\$
59	58	rexbii 2514	. . . 4 $/\$ $/\$
60	53, 59	bitri 184	. . 3 $/\$ $/\$
61	38, 60	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
62	4, 61	exlimddv 1923	1 $/\$ $/\$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 710

wex 1516

wcel 2177

wral 2485

wrex 2486

crab 2489

wss 3170

cop 3641 class class class wbr 4054

cnr 7440

c0r 7441

cr 7954

cltrr 7959

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4170 ax-sep 4173 ax-nul 4181 ax-pow 4229 ax-pr 4264 ax-un 4493 ax-setind 4598 ax-iinf 4649

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-ral 2490 df-rex 2491 df-reu 2492 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-nul 3465 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3860 df-int 3895 df-iun 3938 df-br 4055 df-opab 4117 df-mpt 4118 df-tr 4154 df-eprel 4349 df-id 4353 df-po 4356 df-iso 4357 df-iord 4426 df-on 4428 df-suc 4431 df-iom 4652 df-xp 4694 df-rel 4695 df-cnv 4696 df-co 4697 df-dm 4698 df-rn 4699 df-res 4700 df-ima 4701 df-iota 5246 df-fun 5287 df-fn 5288 df-f 5289 df-f1 5290 df-fo 5291 df-f1o 5292 df-fv 5293 df-ov 5965 df-oprab 5966 df-mpo 5967 df-1st 6244 df-2nd 6245 df-recs 6409 df-irdg 6474 df-1o 6520 df-2o 6521 df-oadd 6524 df-omul 6525 df-er 6638 df-ec 6640 df-qs 6644 df-ni 7447 df-pli 7448 df-mi 7449 df-lti 7450 df-plpq 7487 df-mpq 7488 df-enq 7490 df-nqqs 7491 df-plqqs 7492 df-mqqs 7493 df-1nqqs 7494 df-rq 7495 df-ltnqqs 7496 df-enq0 7567 df-nq0 7568 df-0nq0 7569 df-plq0 7570 df-mq0 7571 df-inp 7609 df-i1p 7610 df-iplp 7611 df-imp 7612 df-iltp 7613 df-enr 7869 df-nr 7870 df-plr 7871 df-mr 7872 df-ltr 7873 df-0r 7874 df-1r 7875 df-m1r 7876 df-r 7965 df-lt 7968

This theorem is referenced by: (None)

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