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Theorem axpre-suploc 8100

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 8131. (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 2290	. . . 4
3	2	cbvexv 1965	. . 3
4	1, 3	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$
5		simplll 533	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
6		simpr 110	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
7		simplrl 535	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$
8		breq2 4087	. . . . . . . 8
9	8	ralbidv 2530	. . . . . . 7
10	9	cbvrexv 2766	. . . . . 6
11		breq1 4086	. . . . . . . 8
12	11	cbvralv 2765	. . . . . . 7
13	12	rexbii 2537	. . . . . 6
14	10, 13	bitri 184	. . . . 5
15	7, 14	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
16		simplrr 536	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
17		breq1 4086	. . . . . . . 8
18		breq1 4086	. . . . . . . . . 10
19	18	rexbidv 2531	. . . . . . . . 9
20	19	orbi1d 796	. . . . . . . 8 $\/$ $\/$
21	17, 20	imbi12d 234	. . . . . . 7 $\/$ $\/$
22		breq2 4087	. . . . . . . 8
23		breq2 4087	. . . . . . . . . 10
24	23	ralbidv 2530	. . . . . . . . 9
25	24	orbi2d 795	. . . . . . . 8 $\/$ $\/$
26	22, 25	imbi12d 234	. . . . . . 7 $\/$ $\/$
27	21, 26	cbvral2v 2778	. . . . . 6 $\/$ $\/$
28		breq2 4087	. . . . . . . . . 10
29	28	cbvrexv 2766	. . . . . . . . 9
30		breq1 4086	. . . . . . . . . 10
31	30	cbvralv 2765	. . . . . . . . 9
32	29, 31	orbi12i 769	. . . . . . . 8 $\/$ $\/$
33	32	imbi2i 226	. . . . . . 7 $\/$ $\/$
34	33	2ralbii 2538	. . . . . 6 $\/$ $\/$
35	27, 34	bitri 184	. . . . 5 $\/$ $\/$
36	16, 35	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
37		eqid 2229	. . . 4
38	5, 6, 15, 36, 37	axpre-suploclemres 8099	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
39	17	notbid 671	. . . . . . . 8
40	39	ralbidv 2530	. . . . . . 7
41	8	imbi1d 231	. . . . . . . 8
42	41	ralbidv 2530	. . . . . . 7
43	40, 42	anbi12d 473	. . . . . 6 $/\$ $/\$
44	43	cbvrexv 2766	. . . . 5 $/\$ $/\$
45	22	notbid 671	. . . . . . . 8
46	45	cbvralv 2765	. . . . . . 7
47		breq1 4086	. . . . . . . . . 10
48	47	rexbidv 2531	. . . . . . . . 9
49	11, 48	imbi12d 234	. . . . . . . 8
50	49	cbvralv 2765	. . . . . . 7
51	46, 50	anbi12i 460	. . . . . 6 $/\$ $/\$
52	51	rexbii 2537	. . . . 5 $/\$ $/\$
53	44, 52	bitri 184	. . . 4 $/\$ $/\$
54		breq2 4087	. . . . . . . . 9
55	54	cbvrexv 2766	. . . . . . . 8
56	55	imbi2i 226	. . . . . . 7
57	56	ralbii 2536	. . . . . 6
58	57	anbi2i 457	. . . . 5 $/\$ $/\$
59	58	rexbii 2537	. . . 4 $/\$ $/\$
60	53, 59	bitri 184	. . 3 $/\$ $/\$
61	38, 60	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
62	4, 61	exlimddv 1945	1 $/\$ $/\$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 713

wex 1538

wcel 2200

wral 2508

wrex 2509

crab 2512

wss 3197

cop 3669 class class class wbr 4083

cnr 7495

c0r 7496

cr 8009

cltrr 8014

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-eprel 4380 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-irdg 6522 df-1o 6568 df-2o 6569 df-oadd 6572 df-omul 6573 df-er 6688 df-ec 6690 df-qs 6694 df-ni 7502 df-pli 7503 df-mi 7504 df-lti 7505 df-plpq 7542 df-mpq 7543 df-enq 7545 df-nqqs 7546 df-plqqs 7547 df-mqqs 7548 df-1nqqs 7549 df-rq 7550 df-ltnqqs 7551 df-enq0 7622 df-nq0 7623 df-0nq0 7624 df-plq0 7625 df-mq0 7626 df-inp 7664 df-i1p 7665 df-iplp 7666 df-imp 7667 df-iltp 7668 df-enr 7924 df-nr 7925 df-plr 7926 df-mr 7927 df-ltr 7928 df-0r 7929 df-1r 7930 df-m1r 7931 df-r 8020 df-lt 8023

This theorem is referenced by: (None)

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