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Mirrors > Home > ILE Home > Th. List > axpre-suploc | Unicode version |
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 7866. (Contributed by Jim Kingdon, 23-Jan-2024.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpre-suploc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 520 | . . 3 | |
2 | eleq1w 2225 | . . . 4 | |
3 | 2 | cbvexv 1905 | . . 3 |
4 | 1, 3 | sylib 121 | . 2 |
5 | simplll 523 | . . . 4 | |
6 | simpr 109 | . . . 4 | |
7 | simplrl 525 | . . . . 5 | |
8 | breq2 3981 | . . . . . . . 8 | |
9 | 8 | ralbidv 2464 | . . . . . . 7 |
10 | 9 | cbvrexv 2691 | . . . . . 6 |
11 | breq1 3980 | . . . . . . . 8 | |
12 | 11 | cbvralv 2690 | . . . . . . 7 |
13 | 12 | rexbii 2471 | . . . . . 6 |
14 | 10, 13 | bitri 183 | . . . . 5 |
15 | 7, 14 | sylibr 133 | . . . 4 |
16 | simplrr 526 | . . . . 5 | |
17 | breq1 3980 | . . . . . . . 8 | |
18 | breq1 3980 | . . . . . . . . . 10 | |
19 | 18 | rexbidv 2465 | . . . . . . . . 9 |
20 | 19 | orbi1d 781 | . . . . . . . 8 |
21 | 17, 20 | imbi12d 233 | . . . . . . 7 |
22 | breq2 3981 | . . . . . . . 8 | |
23 | breq2 3981 | . . . . . . . . . 10 | |
24 | 23 | ralbidv 2464 | . . . . . . . . 9 |
25 | 24 | orbi2d 780 | . . . . . . . 8 |
26 | 22, 25 | imbi12d 233 | . . . . . . 7 |
27 | 21, 26 | cbvral2v 2701 | . . . . . 6 |
28 | breq2 3981 | . . . . . . . . . 10 | |
29 | 28 | cbvrexv 2691 | . . . . . . . . 9 |
30 | breq1 3980 | . . . . . . . . . 10 | |
31 | 30 | cbvralv 2690 | . . . . . . . . 9 |
32 | 29, 31 | orbi12i 754 | . . . . . . . 8 |
33 | 32 | imbi2i 225 | . . . . . . 7 |
34 | 33 | 2ralbii 2472 | . . . . . 6 |
35 | 27, 34 | bitri 183 | . . . . 5 |
36 | 16, 35 | sylibr 133 | . . . 4 |
37 | eqid 2164 | . . . 4 | |
38 | 5, 6, 15, 36, 37 | axpre-suploclemres 7834 | . . 3 |
39 | 17 | notbid 657 | . . . . . . . 8 |
40 | 39 | ralbidv 2464 | . . . . . . 7 |
41 | 8 | imbi1d 230 | . . . . . . . 8 |
42 | 41 | ralbidv 2464 | . . . . . . 7 |
43 | 40, 42 | anbi12d 465 | . . . . . 6 |
44 | 43 | cbvrexv 2691 | . . . . 5 |
45 | 22 | notbid 657 | . . . . . . . 8 |
46 | 45 | cbvralv 2690 | . . . . . . 7 |
47 | breq1 3980 | . . . . . . . . . 10 | |
48 | 47 | rexbidv 2465 | . . . . . . . . 9 |
49 | 11, 48 | imbi12d 233 | . . . . . . . 8 |
50 | 49 | cbvralv 2690 | . . . . . . 7 |
51 | 46, 50 | anbi12i 456 | . . . . . 6 |
52 | 51 | rexbii 2471 | . . . . 5 |
53 | 44, 52 | bitri 183 | . . . 4 |
54 | breq2 3981 | . . . . . . . . 9 | |
55 | 54 | cbvrexv 2691 | . . . . . . . 8 |
56 | 55 | imbi2i 225 | . . . . . . 7 |
57 | 56 | ralbii 2470 | . . . . . 6 |
58 | 57 | anbi2i 453 | . . . . 5 |
59 | 58 | rexbii 2471 | . . . 4 |
60 | 53, 59 | bitri 183 | . . 3 |
61 | 38, 60 | sylib 121 | . 2 |
62 | 4, 61 | exlimddv 1885 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 wex 1479 wcel 2135 wral 2442 wrex 2443 crab 2446 wss 3112 cop 3574 class class class wbr 3977 cnr 7230 c0r 7231 cr 7744 cltrr 7749 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-eprel 4262 df-id 4266 df-po 4269 df-iso 4270 df-iord 4339 df-on 4341 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-recs 6265 df-irdg 6330 df-1o 6376 df-2o 6377 df-oadd 6380 df-omul 6381 df-er 6493 df-ec 6495 df-qs 6499 df-ni 7237 df-pli 7238 df-mi 7239 df-lti 7240 df-plpq 7277 df-mpq 7278 df-enq 7280 df-nqqs 7281 df-plqqs 7282 df-mqqs 7283 df-1nqqs 7284 df-rq 7285 df-ltnqqs 7286 df-enq0 7357 df-nq0 7358 df-0nq0 7359 df-plq0 7360 df-mq0 7361 df-inp 7399 df-i1p 7400 df-iplp 7401 df-imp 7402 df-iltp 7403 df-enr 7659 df-nr 7660 df-plr 7661 df-mr 7662 df-ltr 7663 df-0r 7664 df-1r 7665 df-m1r 7666 df-r 7755 df-lt 7758 |
This theorem is referenced by: (None) |
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