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Theorem axpre-suploc 8085

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 8116. (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 4086	. . . . . . . 8
9	8	ralbidv 2530	. . . . . . 7
10	9	cbvrexv 2766	. . . . . 6
11		breq1 4085	. . . . . . . 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 4085	. . . . . . . 8
18		breq1 4085	. . . . . . . . . 10
19	18	rexbidv 2531	. . . . . . . . 9
20	19	orbi1d 796	. . . . . . . 8 $\/$ $\/$
21	17, 20	imbi12d 234	. . . . . . 7 $\/$ $\/$
22		breq2 4086	. . . . . . . 8
23		breq2 4086	. . . . . . . . . 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 4086	. . . . . . . . . 10
29	28	cbvrexv 2766	. . . . . . . . 9
30		breq1 4085	. . . . . . . . . 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 8084	. . 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 4085	. . . . . . . . . 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 4086	. . . . . . . . 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 4082

cnr 7480

c0r 7481

cr 7994

cltrr 7999

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 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679

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 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-eprel 4379 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-irdg 6514 df-1o 6560 df-2o 6561 df-oadd 6564 df-omul 6565 df-er 6678 df-ec 6680 df-qs 6684 df-ni 7487 df-pli 7488 df-mi 7489 df-lti 7490 df-plpq 7527 df-mpq 7528 df-enq 7530 df-nqqs 7531 df-plqqs 7532 df-mqqs 7533 df-1nqqs 7534 df-rq 7535 df-ltnqqs 7536 df-enq0 7607 df-nq0 7608 df-0nq0 7609 df-plq0 7610 df-mq0 7611 df-inp 7649 df-i1p 7650 df-iplp 7651 df-imp 7652 df-iltp 7653 df-enr 7909 df-nr 7910 df-plr 7911 df-mr 7912 df-ltr 7913 df-0r 7914 df-1r 7915 df-m1r 7916 df-r 8005 df-lt 8008

This theorem is referenced by: (None)

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