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 7864

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 7895. (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 525	. . 3 $/\$ $/\$ $/\$ $\/$
2		eleq1w 2231	. . . 4
3	2	cbvexv 1911	. . 3
4	1, 3	sylib 121	. 2 $/\$ $/\$ $/\$ $\/$
5		simplll 528	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
6		simpr 109	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
7		simplrl 530	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$
8		breq2 3993	. . . . . . . 8
9	8	ralbidv 2470	. . . . . . 7
10	9	cbvrexv 2697	. . . . . 6
11		breq1 3992	. . . . . . . 8
12	11	cbvralv 2696	. . . . . . 7
13	12	rexbii 2477	. . . . . 6
14	10, 13	bitri 183	. . . . 5
15	7, 14	sylibr 133	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
16		simplrr 531	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
17		breq1 3992	. . . . . . . 8
18		breq1 3992	. . . . . . . . . 10
19	18	rexbidv 2471	. . . . . . . . 9
20	19	orbi1d 786	. . . . . . . 8 $\/$ $\/$
21	17, 20	imbi12d 233	. . . . . . 7 $\/$ $\/$
22		breq2 3993	. . . . . . . 8
23		breq2 3993	. . . . . . . . . 10
24	23	ralbidv 2470	. . . . . . . . 9
25	24	orbi2d 785	. . . . . . . 8 $\/$ $\/$
26	22, 25	imbi12d 233	. . . . . . 7 $\/$ $\/$
27	21, 26	cbvral2v 2709	. . . . . 6 $\/$ $\/$
28		breq2 3993	. . . . . . . . . 10
29	28	cbvrexv 2697	. . . . . . . . 9
30		breq1 3992	. . . . . . . . . 10
31	30	cbvralv 2696	. . . . . . . . 9
32	29, 31	orbi12i 759	. . . . . . . 8 $\/$ $\/$
33	32	imbi2i 225	. . . . . . 7 $\/$ $\/$
34	33	2ralbii 2478	. . . . . 6 $\/$ $\/$
35	27, 34	bitri 183	. . . . 5 $\/$ $\/$
36	16, 35	sylibr 133	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
37		eqid 2170	. . . 4
38	5, 6, 15, 36, 37	axpre-suploclemres 7863	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
39	17	notbid 662	. . . . . . . 8
40	39	ralbidv 2470	. . . . . . 7
41	8	imbi1d 230	. . . . . . . 8
42	41	ralbidv 2470	. . . . . . 7
43	40, 42	anbi12d 470	. . . . . 6 $/\$ $/\$
44	43	cbvrexv 2697	. . . . 5 $/\$ $/\$
45	22	notbid 662	. . . . . . . 8
46	45	cbvralv 2696	. . . . . . 7
47		breq1 3992	. . . . . . . . . 10
48	47	rexbidv 2471	. . . . . . . . 9
49	11, 48	imbi12d 233	. . . . . . . 8
50	49	cbvralv 2696	. . . . . . 7
51	46, 50	anbi12i 457	. . . . . 6 $/\$ $/\$
52	51	rexbii 2477	. . . . 5 $/\$ $/\$
53	44, 52	bitri 183	. . . 4 $/\$ $/\$
54		breq2 3993	. . . . . . . . 9
55	54	cbvrexv 2697	. . . . . . . 8
56	55	imbi2i 225	. . . . . . 7
57	56	ralbii 2476	. . . . . 6
58	57	anbi2i 454	. . . . 5 $/\$ $/\$
59	58	rexbii 2477	. . . 4 $/\$ $/\$
60	53, 59	bitri 183	. . 3 $/\$ $/\$
61	38, 60	sylib 121	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
62	4, 61	exlimddv 1891	1 $/\$ $/\$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103 $\/$ wo 703

wex 1485

wcel 2141

wral 2448

wrex 2449

crab 2452

wss 3121

cop 3586 class class class wbr 3989

cnr 7259

c0r 7260

cr 7773

cltrr 7778

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-eprel 4274 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-1o 6395 df-2o 6396 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-pli 7267 df-mi 7268 df-lti 7269 df-plpq 7306 df-mpq 7307 df-enq 7309 df-nqqs 7310 df-plqqs 7311 df-mqqs 7312 df-1nqqs 7313 df-rq 7314 df-ltnqqs 7315 df-enq0 7386 df-nq0 7387 df-0nq0 7388 df-plq0 7389 df-mq0 7390 df-inp 7428 df-i1p 7429 df-iplp 7430 df-imp 7431 df-iltp 7432 df-enr 7688 df-nr 7689 df-plr 7690 df-mr 7691 df-ltr 7692 df-0r 7693 df-1r 7694 df-m1r 7695 df-r 7784 df-lt 7787

This theorem is referenced by: (None)

W3C validator