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Mirrors > Home > ILE Home > Th. List > axsuploc | Unicode version |
Description: An inhabited, bounded-above, located set of reals has a supremum. Axiom for real and complex numbers, derived from ZF set theory. (This restates ax-pre-suploc 7865 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.) |
Ref | Expression |
---|---|
axsuploc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel2 3132 | . . . . . . . . . 10 | |
2 | ltxrlt 7955 | . . . . . . . . . 10 | |
3 | 1, 2 | sylan 281 | . . . . . . . . 9 |
4 | 3 | an32s 558 | . . . . . . . 8 |
5 | 4 | ralbidva 2460 | . . . . . . 7 |
6 | 5 | rexbidva 2461 | . . . . . 6 |
7 | simplr 520 | . . . . . . . . . 10 | |
8 | simpr 109 | . . . . . . . . . 10 | |
9 | ltxrlt 7955 | . . . . . . . . . 10 | |
10 | 7, 8, 9 | syl2anc 409 | . . . . . . . . 9 |
11 | simpllr 524 | . . . . . . . . . . . 12 | |
12 | ssel2 3132 | . . . . . . . . . . . . . 14 | |
13 | 12 | adantlr 469 | . . . . . . . . . . . . 13 |
14 | 13 | adantlr 469 | . . . . . . . . . . . 12 |
15 | ltxrlt 7955 | . . . . . . . . . . . 12 | |
16 | 11, 14, 15 | syl2anc 409 | . . . . . . . . . . 11 |
17 | 16 | rexbidva 2461 | . . . . . . . . . 10 |
18 | simplr 520 | . . . . . . . . . . . 12 | |
19 | ltxrlt 7955 | . . . . . . . . . . . 12 | |
20 | 14, 18, 19 | syl2anc 409 | . . . . . . . . . . 11 |
21 | 20 | ralbidva 2460 | . . . . . . . . . 10 |
22 | 17, 21 | orbi12d 783 | . . . . . . . . 9 |
23 | 10, 22 | imbi12d 233 | . . . . . . . 8 |
24 | 23 | ralbidva 2460 | . . . . . . 7 |
25 | 24 | ralbidva 2460 | . . . . . 6 |
26 | 6, 25 | anbi12d 465 | . . . . 5 |
27 | 26 | adantr 274 | . . . 4 |
28 | 27 | pm5.32i 450 | . . 3 |
29 | ax-pre-suploc 7865 | . . 3 | |
30 | 28, 29 | sylbi 120 | . 2 |
31 | simplr 520 | . . . . . . . . 9 | |
32 | 1 | adantlr 469 | . . . . . . . . 9 |
33 | 31, 32, 9 | syl2anc 409 | . . . . . . . 8 |
34 | 33 | bicomd 140 | . . . . . . 7 |
35 | 34 | notbid 657 | . . . . . 6 |
36 | 35 | ralbidva 2460 | . . . . 5 |
37 | 8, 7, 2 | syl2anc 409 | . . . . . . . 8 |
38 | 37 | bicomd 140 | . . . . . . 7 |
39 | ltxrlt 7955 | . . . . . . . . . 10 | |
40 | 18, 14, 39 | syl2anc 409 | . . . . . . . . 9 |
41 | 40 | bicomd 140 | . . . . . . . 8 |
42 | 41 | rexbidva 2461 | . . . . . . 7 |
43 | 38, 42 | imbi12d 233 | . . . . . 6 |
44 | 43 | ralbidva 2460 | . . . . 5 |
45 | 36, 44 | anbi12d 465 | . . . 4 |
46 | 45 | rexbidva 2461 | . . 3 |
47 | 46 | ad2antrr 480 | . 2 |
48 | 30, 47 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 wex 1479 wcel 2135 wral 2442 wrex 2443 wss 3111 class class class wbr 3976 cr 7743 cltrr 7748 clt 7924 |
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-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-pre-suploc 7865 |
This theorem depends on definitions: df-bi 116 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-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-pnf 7926 df-mnf 7927 df-ltxr 7929 |
This theorem is referenced by: dedekindeulemlub 13139 suplociccreex 13143 |
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