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 7901

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 7932. (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 2238	. . . 4
3	2	cbvexv 1918	. . 3
4	1, 3	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$
5		simplll 533	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
6		simpr 110	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
7		simplrl 535	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$
8		breq2 4008	. . . . . . . 8
9	8	ralbidv 2477	. . . . . . 7
10	9	cbvrexv 2705	. . . . . 6
11		breq1 4007	. . . . . . . 8
12	11	cbvralv 2704	. . . . . . 7
13	12	rexbii 2484	. . . . . 6
14	10, 13	bitri 184	. . . . 5
15	7, 14	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
16		simplrr 536	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
17		breq1 4007	. . . . . . . 8
18		breq1 4007	. . . . . . . . . 10
19	18	rexbidv 2478	. . . . . . . . 9
20	19	orbi1d 791	. . . . . . . 8 $\/$ $\/$
21	17, 20	imbi12d 234	. . . . . . 7 $\/$ $\/$
22		breq2 4008	. . . . . . . 8
23		breq2 4008	. . . . . . . . . 10
24	23	ralbidv 2477	. . . . . . . . 9
25	24	orbi2d 790	. . . . . . . 8 $\/$ $\/$
26	22, 25	imbi12d 234	. . . . . . 7 $\/$ $\/$
27	21, 26	cbvral2v 2717	. . . . . 6 $\/$ $\/$
28		breq2 4008	. . . . . . . . . 10
29	28	cbvrexv 2705	. . . . . . . . 9
30		breq1 4007	. . . . . . . . . 10
31	30	cbvralv 2704	. . . . . . . . 9
32	29, 31	orbi12i 764	. . . . . . . 8 $\/$ $\/$
33	32	imbi2i 226	. . . . . . 7 $\/$ $\/$
34	33	2ralbii 2485	. . . . . 6 $\/$ $\/$
35	27, 34	bitri 184	. . . . 5 $\/$ $\/$
36	16, 35	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
37		eqid 2177	. . . 4
38	5, 6, 15, 36, 37	axpre-suploclemres 7900	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
39	17	notbid 667	. . . . . . . 8
40	39	ralbidv 2477	. . . . . . 7
41	8	imbi1d 231	. . . . . . . 8
42	41	ralbidv 2477	. . . . . . 7
43	40, 42	anbi12d 473	. . . . . 6 $/\$ $/\$
44	43	cbvrexv 2705	. . . . 5 $/\$ $/\$
45	22	notbid 667	. . . . . . . 8
46	45	cbvralv 2704	. . . . . . 7
47		breq1 4007	. . . . . . . . . 10
48	47	rexbidv 2478	. . . . . . . . 9
49	11, 48	imbi12d 234	. . . . . . . 8
50	49	cbvralv 2704	. . . . . . 7
51	46, 50	anbi12i 460	. . . . . 6 $/\$ $/\$
52	51	rexbii 2484	. . . . 5 $/\$ $/\$
53	44, 52	bitri 184	. . . 4 $/\$ $/\$
54		breq2 4008	. . . . . . . . 9
55	54	cbvrexv 2705	. . . . . . . 8
56	55	imbi2i 226	. . . . . . 7
57	56	ralbii 2483	. . . . . 6
58	57	anbi2i 457	. . . . 5 $/\$ $/\$
59	58	rexbii 2484	. . . 4 $/\$ $/\$
60	53, 59	bitri 184	. . 3 $/\$ $/\$
61	38, 60	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
62	4, 61	exlimddv 1898	1 $/\$ $/\$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 708

wex 1492

wcel 2148

wral 2455

wrex 2456

crab 2459

wss 3130

cop 3596 class class class wbr 4004

cnr 7296

c0r 7297

cr 7810

cltrr 7815

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-eprel 4290 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-irdg 6371 df-1o 6417 df-2o 6418 df-oadd 6421 df-omul 6422 df-er 6535 df-ec 6537 df-qs 6541 df-ni 7303 df-pli 7304 df-mi 7305 df-lti 7306 df-plpq 7343 df-mpq 7344 df-enq 7346 df-nqqs 7347 df-plqqs 7348 df-mqqs 7349 df-1nqqs 7350 df-rq 7351 df-ltnqqs 7352 df-enq0 7423 df-nq0 7424 df-0nq0 7425 df-plq0 7426 df-mq0 7427 df-inp 7465 df-i1p 7466 df-iplp 7467 df-imp 7468 df-iltp 7469 df-enr 7725 df-nr 7726 df-plr 7727 df-mr 7728 df-ltr 7729 df-0r 7730 df-1r 7731 df-m1r 7732 df-r 7821 df-lt 7824

This theorem is referenced by: (None)

W3C validator