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 7986

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 8017. (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 2257	. . . 4
3	2	cbvexv 1933	. . 3
4	1, 3	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$
5		simplll 533	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
6		simpr 110	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
7		simplrl 535	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$
8		breq2 4038	. . . . . . . 8
9	8	ralbidv 2497	. . . . . . 7
10	9	cbvrexv 2730	. . . . . 6
11		breq1 4037	. . . . . . . 8
12	11	cbvralv 2729	. . . . . . 7
13	12	rexbii 2504	. . . . . 6
14	10, 13	bitri 184	. . . . 5
15	7, 14	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
16		simplrr 536	. . . . 5 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
17		breq1 4037	. . . . . . . 8
18		breq1 4037	. . . . . . . . . 10
19	18	rexbidv 2498	. . . . . . . . 9
20	19	orbi1d 792	. . . . . . . 8 $\/$ $\/$
21	17, 20	imbi12d 234	. . . . . . 7 $\/$ $\/$
22		breq2 4038	. . . . . . . 8
23		breq2 4038	. . . . . . . . . 10
24	23	ralbidv 2497	. . . . . . . . 9
25	24	orbi2d 791	. . . . . . . 8 $\/$ $\/$
26	22, 25	imbi12d 234	. . . . . . 7 $\/$ $\/$
27	21, 26	cbvral2v 2742	. . . . . 6 $\/$ $\/$
28		breq2 4038	. . . . . . . . . 10
29	28	cbvrexv 2730	. . . . . . . . 9
30		breq1 4037	. . . . . . . . . 10
31	30	cbvralv 2729	. . . . . . . . 9
32	29, 31	orbi12i 765	. . . . . . . 8 $\/$ $\/$
33	32	imbi2i 226	. . . . . . 7 $\/$ $\/$
34	33	2ralbii 2505	. . . . . 6 $\/$ $\/$
35	27, 34	bitri 184	. . . . 5 $\/$ $\/$
36	16, 35	sylibr 134	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
37		eqid 2196	. . . 4
38	5, 6, 15, 36, 37	axpre-suploclemres 7985	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
39	17	notbid 668	. . . . . . . 8
40	39	ralbidv 2497	. . . . . . 7
41	8	imbi1d 231	. . . . . . . 8
42	41	ralbidv 2497	. . . . . . 7
43	40, 42	anbi12d 473	. . . . . 6 $/\$ $/\$
44	43	cbvrexv 2730	. . . . 5 $/\$ $/\$
45	22	notbid 668	. . . . . . . 8
46	45	cbvralv 2729	. . . . . . 7
47		breq1 4037	. . . . . . . . . 10
48	47	rexbidv 2498	. . . . . . . . 9
49	11, 48	imbi12d 234	. . . . . . . 8
50	49	cbvralv 2729	. . . . . . 7
51	46, 50	anbi12i 460	. . . . . 6 $/\$ $/\$
52	51	rexbii 2504	. . . . 5 $/\$ $/\$
53	44, 52	bitri 184	. . . 4 $/\$ $/\$
54		breq2 4038	. . . . . . . . 9
55	54	cbvrexv 2730	. . . . . . . 8
56	55	imbi2i 226	. . . . . . 7
57	56	ralbii 2503	. . . . . 6
58	57	anbi2i 457	. . . . 5 $/\$ $/\$
59	58	rexbii 2504	. . . 4 $/\$ $/\$
60	53, 59	bitri 184	. . 3 $/\$ $/\$
61	38, 60	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
62	4, 61	exlimddv 1913	1 $/\$ $/\$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

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

wex 1506

wcel 2167

wral 2475

wrex 2476

crab 2479

wss 3157

cop 3626 class class class wbr 4034

cnr 7381

c0r 7382

cr 7895

cltrr 7900

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-eprel 4325 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-1o 6483 df-2o 6484 df-oadd 6487 df-omul 6488 df-er 6601 df-ec 6603 df-qs 6607 df-ni 7388 df-pli 7389 df-mi 7390 df-lti 7391 df-plpq 7428 df-mpq 7429 df-enq 7431 df-nqqs 7432 df-plqqs 7433 df-mqqs 7434 df-1nqqs 7435 df-rq 7436 df-ltnqqs 7437 df-enq0 7508 df-nq0 7509 df-0nq0 7510 df-plq0 7511 df-mq0 7512 df-inp 7550 df-i1p 7551 df-iplp 7552 df-imp 7553 df-iltp 7554 df-enr 7810 df-nr 7811 df-plr 7812 df-mr 7813 df-ltr 7814 df-0r 7815 df-1r 7816 df-m1r 7817 df-r 7906 df-lt 7909

This theorem is referenced by: (None)

W3C validator