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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 7741 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.) |
Ref | Expression |
---|---|
axsuploc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel2 3092 | . . . . . . . . . 10 | |
2 | ltxrlt 7830 | . . . . . . . . . 10 | |
3 | 1, 2 | sylan 281 | . . . . . . . . 9 |
4 | 3 | an32s 557 | . . . . . . . 8 |
5 | 4 | ralbidva 2433 | . . . . . . 7 |
6 | 5 | rexbidva 2434 | . . . . . 6 |
7 | simplr 519 | . . . . . . . . . 10 | |
8 | simpr 109 | . . . . . . . . . 10 | |
9 | ltxrlt 7830 | . . . . . . . . . 10 | |
10 | 7, 8, 9 | syl2anc 408 | . . . . . . . . 9 |
11 | simpllr 523 | . . . . . . . . . . . 12 | |
12 | ssel2 3092 | . . . . . . . . . . . . . 14 | |
13 | 12 | adantlr 468 | . . . . . . . . . . . . 13 |
14 | 13 | adantlr 468 | . . . . . . . . . . . 12 |
15 | ltxrlt 7830 | . . . . . . . . . . . 12 | |
16 | 11, 14, 15 | syl2anc 408 | . . . . . . . . . . 11 |
17 | 16 | rexbidva 2434 | . . . . . . . . . 10 |
18 | simplr 519 | . . . . . . . . . . . 12 | |
19 | ltxrlt 7830 | . . . . . . . . . . . 12 | |
20 | 14, 18, 19 | syl2anc 408 | . . . . . . . . . . 11 |
21 | 20 | ralbidva 2433 | . . . . . . . . . 10 |
22 | 17, 21 | orbi12d 782 | . . . . . . . . 9 |
23 | 10, 22 | imbi12d 233 | . . . . . . . 8 |
24 | 23 | ralbidva 2433 | . . . . . . 7 |
25 | 24 | ralbidva 2433 | . . . . . 6 |
26 | 6, 25 | anbi12d 464 | . . . . 5 |
27 | 26 | adantr 274 | . . . 4 |
28 | 27 | pm5.32i 449 | . . 3 |
29 | ax-pre-suploc 7741 | . . 3 | |
30 | 28, 29 | sylbi 120 | . 2 |
31 | simplr 519 | . . . . . . . . 9 | |
32 | 1 | adantlr 468 | . . . . . . . . 9 |
33 | 31, 32, 9 | syl2anc 408 | . . . . . . . 8 |
34 | 33 | bicomd 140 | . . . . . . 7 |
35 | 34 | notbid 656 | . . . . . 6 |
36 | 35 | ralbidva 2433 | . . . . 5 |
37 | 8, 7, 2 | syl2anc 408 | . . . . . . . 8 |
38 | 37 | bicomd 140 | . . . . . . 7 |
39 | ltxrlt 7830 | . . . . . . . . . 10 | |
40 | 18, 14, 39 | syl2anc 408 | . . . . . . . . 9 |
41 | 40 | bicomd 140 | . . . . . . . 8 |
42 | 41 | rexbidva 2434 | . . . . . . 7 |
43 | 38, 42 | imbi12d 233 | . . . . . 6 |
44 | 43 | ralbidva 2433 | . . . . 5 |
45 | 36, 44 | anbi12d 464 | . . . 4 |
46 | 45 | rexbidva 2434 | . . 3 |
47 | 46 | ad2antrr 479 | . 2 |
48 | 30, 47 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 wex 1468 wcel 1480 wral 2416 wrex 2417 wss 3071 class class class wbr 3929 cr 7619 cltrr 7624 clt 7800 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-suploc 7741 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-pnf 7802 df-mnf 7803 df-ltxr 7805 |
This theorem is referenced by: dedekindeulemlub 12767 suplociccreex 12771 |
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