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Theorem axpre-suploc 8014

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 8045. (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 2265	. . . 4
3	2	cbvexv 1941	. . 3
4	1, 3	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$
5		simplll 533	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
6		simpr 110	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
7		simplrl 535	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$
8		breq2 4047	. . . . . . . 8
9	8	ralbidv 2505	. . . . . . 7
10	9	cbvrexv 2738	. . . . . 6
11		breq1 4046	. . . . . . . 8
12	11	cbvralv 2737	. . . . . . 7
13	12	rexbii 2512	. . . . . 6
14	10, 13	bitri 184	. . . . 5
15	7, 14	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
16		simplrr 536	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
17		breq1 4046	. . . . . . . 8
18		breq1 4046	. . . . . . . . . 10
19	18	rexbidv 2506	. . . . . . . . 9
20	19	orbi1d 792	. . . . . . . 8 $\/$ $\/$
21	17, 20	imbi12d 234	. . . . . . 7 $\/$ $\/$
22		breq2 4047	. . . . . . . 8
23		breq2 4047	. . . . . . . . . 10
24	23	ralbidv 2505	. . . . . . . . 9
25	24	orbi2d 791	. . . . . . . 8 $\/$ $\/$
26	22, 25	imbi12d 234	. . . . . . 7 $\/$ $\/$
27	21, 26	cbvral2v 2750	. . . . . 6 $\/$ $\/$
28		breq2 4047	. . . . . . . . . 10
29	28	cbvrexv 2738	. . . . . . . . 9
30		breq1 4046	. . . . . . . . . 10
31	30	cbvralv 2737	. . . . . . . . 9
32	29, 31	orbi12i 765	. . . . . . . 8 $\/$ $\/$
33	32	imbi2i 226	. . . . . . 7 $\/$ $\/$
34	33	2ralbii 2513	. . . . . 6 $\/$ $\/$
35	27, 34	bitri 184	. . . . 5 $\/$ $\/$
36	16, 35	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
37		eqid 2204	. . . 4
38	5, 6, 15, 36, 37	axpre-suploclemres 8013	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
39	17	notbid 668	. . . . . . . 8
40	39	ralbidv 2505	. . . . . . 7
41	8	imbi1d 231	. . . . . . . 8
42	41	ralbidv 2505	. . . . . . 7
43	40, 42	anbi12d 473	. . . . . 6 $/\$ $/\$
44	43	cbvrexv 2738	. . . . 5 $/\$ $/\$
45	22	notbid 668	. . . . . . . 8
46	45	cbvralv 2737	. . . . . . 7
47		breq1 4046	. . . . . . . . . 10
48	47	rexbidv 2506	. . . . . . . . 9
49	11, 48	imbi12d 234	. . . . . . . 8
50	49	cbvralv 2737	. . . . . . 7
51	46, 50	anbi12i 460	. . . . . 6 $/\$ $/\$
52	51	rexbii 2512	. . . . 5 $/\$ $/\$
53	44, 52	bitri 184	. . . 4 $/\$ $/\$
54		breq2 4047	. . . . . . . . 9
55	54	cbvrexv 2738	. . . . . . . 8
56	55	imbi2i 226	. . . . . . 7
57	56	ralbii 2511	. . . . . 6
58	57	anbi2i 457	. . . . 5 $/\$ $/\$
59	58	rexbii 2512	. . . 4 $/\$ $/\$
60	53, 59	bitri 184	. . 3 $/\$ $/\$
61	38, 60	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
62	4, 61	exlimddv 1921	1 $/\$ $/\$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 709

wex 1514

wcel 2175

wral 2483

wrex 2484

crab 2487

wss 3165

cop 3635 class class class wbr 4043

cnr 7409

c0r 7410

cr 7923

cltrr 7928

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 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-iinf 4635

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-eprel 4335 df-id 4339 df-po 4342 df-iso 4343 df-iord 4412 df-on 4414 df-suc 4417 df-iom 4638 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-ov 5946 df-oprab 5947 df-mpo 5948 df-1st 6225 df-2nd 6226 df-recs 6390 df-irdg 6455 df-1o 6501 df-2o 6502 df-oadd 6505 df-omul 6506 df-er 6619 df-ec 6621 df-qs 6625 df-ni 7416 df-pli 7417 df-mi 7418 df-lti 7419 df-plpq 7456 df-mpq 7457 df-enq 7459 df-nqqs 7460 df-plqqs 7461 df-mqqs 7462 df-1nqqs 7463 df-rq 7464 df-ltnqqs 7465 df-enq0 7536 df-nq0 7537 df-0nq0 7538 df-plq0 7539 df-mq0 7540 df-inp 7578 df-i1p 7579 df-iplp 7580 df-imp 7581 df-iltp 7582 df-enr 7838 df-nr 7839 df-plr 7840 df-mr 7841 df-ltr 7842 df-0r 7843 df-1r 7844 df-m1r 7845 df-r 7934 df-lt 7937

This theorem is referenced by: (None)

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