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Theorem axpre-suploc 8233

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 8264. (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 2295	. . . 4
3	2	cbvexv 1970	. . 3
4	1, 3	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$
5		simplll 535	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
6		simpr 110	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
7		simplrl 537	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$
8		breq2 4118	. . . . . . . 8
9	8	ralbidv 2544	. . . . . . 7
10	9	cbvrexv 2781	. . . . . 6
11		breq1 4117	. . . . . . . 8
12	11	cbvralv 2780	. . . . . . 7
13	12	rexbii 2551	. . . . . 6
14	10, 13	bitri 184	. . . . 5
15	7, 14	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
16		simplrr 538	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
17		breq1 4117	. . . . . . . 8
18		breq1 4117	. . . . . . . . . 10
19	18	rexbidv 2545	. . . . . . . . 9
20	19	orbi1d 799	. . . . . . . 8 $\/$ $\/$
21	17, 20	imbi12d 234	. . . . . . 7 $\/$ $\/$
22		breq2 4118	. . . . . . . 8
23		breq2 4118	. . . . . . . . . 10
24	23	ralbidv 2544	. . . . . . . . 9
25	24	orbi2d 798	. . . . . . . 8 $\/$ $\/$
26	22, 25	imbi12d 234	. . . . . . 7 $\/$ $\/$
27	21, 26	cbvral2v 2793	. . . . . 6 $\/$ $\/$
28		breq2 4118	. . . . . . . . . 10
29	28	cbvrexv 2781	. . . . . . . . 9
30		breq1 4117	. . . . . . . . . 10
31	30	cbvralv 2780	. . . . . . . . 9
32	29, 31	orbi12i 772	. . . . . . . 8 $\/$ $\/$
33	32	imbi2i 226	. . . . . . 7 $\/$ $\/$
34	33	2ralbii 2552	. . . . . 6 $\/$ $\/$
35	27, 34	bitri 184	. . . . 5 $\/$ $\/$
36	16, 35	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
37		eqid 2234	. . . 4
38	5, 6, 15, 36, 37	axpre-suploclemres 8232	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
39	17	notbid 673	. . . . . . . 8
40	39	ralbidv 2544	. . . . . . 7
41	8	imbi1d 231	. . . . . . . 8
42	41	ralbidv 2544	. . . . . . 7
43	40, 42	anbi12d 473	. . . . . 6 $/\$ $/\$
44	43	cbvrexv 2781	. . . . 5 $/\$ $/\$
45	22	notbid 673	. . . . . . . 8
46	45	cbvralv 2780	. . . . . . 7
47		breq1 4117	. . . . . . . . . 10
48	47	rexbidv 2545	. . . . . . . . 9
49	11, 48	imbi12d 234	. . . . . . . 8
50	49	cbvralv 2780	. . . . . . 7
51	46, 50	anbi12i 460	. . . . . 6 $/\$ $/\$
52	51	rexbii 2551	. . . . 5 $/\$ $/\$
53	44, 52	bitri 184	. . . 4 $/\$ $/\$
54		breq2 4118	. . . . . . . . 9
55	54	cbvrexv 2781	. . . . . . . 8
56	55	imbi2i 226	. . . . . . 7
57	56	ralbii 2550	. . . . . 6
58	57	anbi2i 457	. . . . 5 $/\$ $/\$
59	58	rexbii 2551	. . . 4 $/\$ $/\$
60	53, 59	bitri 184	. . 3 $/\$ $/\$
61	38, 60	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
62	4, 61	exlimddv 1950	1 $/\$ $/\$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

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

wex 1541

wcel 2205

wral 2522

wrex 2523

crab 2526

wss 3214

cop 3697 class class class wbr 4114

cnr 7628

c0r 7629

cr 8142

cltrr 8147

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 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715

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 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-eprel 4415 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-1o 6660 df-2o 6661 df-oadd 6664 df-omul 6665 df-er 6780 df-ec 6782 df-qs 6786 df-ni 7635 df-pli 7636 df-mi 7637 df-lti 7638 df-plpq 7675 df-mpq 7676 df-enq 7678 df-nqqs 7679 df-plqqs 7680 df-mqqs 7681 df-1nqqs 7682 df-rq 7683 df-ltnqqs 7684 df-enq0 7755 df-nq0 7756 df-0nq0 7757 df-plq0 7758 df-mq0 7759 df-inp 7797 df-i1p 7798 df-iplp 7799 df-imp 7800 df-iltp 7801 df-enr 8057 df-nr 8058 df-plr 8059 df-mr 8060 df-ltr 8061 df-0r 8062 df-1r 8063 df-m1r 8064 df-r 8153 df-lt 8156

This theorem is referenced by: (None)

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