axpre-suploc - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > axpre-suploc		Unicode version

Theorem axpre-suploc 7964

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 7995. (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 2254	. . . 4
3	2	cbvexv 1930	. . 3
4	1, 3	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$
5		simplll 533	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
6		simpr 110	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
7		simplrl 535	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$
8		breq2 4034	. . . . . . . 8
9	8	ralbidv 2494	. . . . . . 7
10	9	cbvrexv 2727	. . . . . 6
11		breq1 4033	. . . . . . . 8
12	11	cbvralv 2726	. . . . . . 7
13	12	rexbii 2501	. . . . . 6
14	10, 13	bitri 184	. . . . 5
15	7, 14	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
16		simplrr 536	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
17		breq1 4033	. . . . . . . 8
18		breq1 4033	. . . . . . . . . 10
19	18	rexbidv 2495	. . . . . . . . 9
20	19	orbi1d 792	. . . . . . . 8 $\/$ $\/$
21	17, 20	imbi12d 234	. . . . . . 7 $\/$ $\/$
22		breq2 4034	. . . . . . . 8
23		breq2 4034	. . . . . . . . . 10
24	23	ralbidv 2494	. . . . . . . . 9
25	24	orbi2d 791	. . . . . . . 8 $\/$ $\/$
26	22, 25	imbi12d 234	. . . . . . 7 $\/$ $\/$
27	21, 26	cbvral2v 2739	. . . . . 6 $\/$ $\/$
28		breq2 4034	. . . . . . . . . 10
29	28	cbvrexv 2727	. . . . . . . . 9
30		breq1 4033	. . . . . . . . . 10
31	30	cbvralv 2726	. . . . . . . . 9
32	29, 31	orbi12i 765	. . . . . . . 8 $\/$ $\/$
33	32	imbi2i 226	. . . . . . 7 $\/$ $\/$
34	33	2ralbii 2502	. . . . . 6 $\/$ $\/$
35	27, 34	bitri 184	. . . . 5 $\/$ $\/$
36	16, 35	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
37		eqid 2193	. . . 4
38	5, 6, 15, 36, 37	axpre-suploclemres 7963	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
39	17	notbid 668	. . . . . . . 8
40	39	ralbidv 2494	. . . . . . 7
41	8	imbi1d 231	. . . . . . . 8
42	41	ralbidv 2494	. . . . . . 7
43	40, 42	anbi12d 473	. . . . . 6 $/\$ $/\$
44	43	cbvrexv 2727	. . . . 5 $/\$ $/\$
45	22	notbid 668	. . . . . . . 8
46	45	cbvralv 2726	. . . . . . 7
47		breq1 4033	. . . . . . . . . 10
48	47	rexbidv 2495	. . . . . . . . 9
49	11, 48	imbi12d 234	. . . . . . . 8
50	49	cbvralv 2726	. . . . . . 7
51	46, 50	anbi12i 460	. . . . . 6 $/\$ $/\$
52	51	rexbii 2501	. . . . 5 $/\$ $/\$
53	44, 52	bitri 184	. . . 4 $/\$ $/\$
54		breq2 4034	. . . . . . . . 9
55	54	cbvrexv 2727	. . . . . . . 8
56	55	imbi2i 226	. . . . . . 7
57	56	ralbii 2500	. . . . . 6
58	57	anbi2i 457	. . . . 5 $/\$ $/\$
59	58	rexbii 2501	. . . 4 $/\$ $/\$
60	53, 59	bitri 184	. . 3 $/\$ $/\$
61	38, 60	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
62	4, 61	exlimddv 1910	1 $/\$ $/\$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

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

wex 1503

wcel 2164

wral 2472

wrex 2473

crab 2476

wss 3154

cop 3622 class class class wbr 4030

cnr 7359

c0r 7360

cr 7873

cltrr 7878

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-eprel 4321 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-irdg 6425 df-1o 6471 df-2o 6472 df-oadd 6475 df-omul 6476 df-er 6589 df-ec 6591 df-qs 6595 df-ni 7366 df-pli 7367 df-mi 7368 df-lti 7369 df-plpq 7406 df-mpq 7407 df-enq 7409 df-nqqs 7410 df-plqqs 7411 df-mqqs 7412 df-1nqqs 7413 df-rq 7414 df-ltnqqs 7415 df-enq0 7486 df-nq0 7487 df-0nq0 7488 df-plq0 7489 df-mq0 7490 df-inp 7528 df-i1p 7529 df-iplp 7530 df-imp 7531 df-iltp 7532 df-enr 7788 df-nr 7789 df-plr 7790 df-mr 7791 df-ltr 7792 df-0r 7793 df-1r 7794 df-m1r 7795 df-r 7884 df-lt 7887

This theorem is referenced by: (None)

W3C validator