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Mirrors > Home > ILE Home > Th. List > infregelbex | Unicode version |
Description: Any lower bound of a set of real numbers with an infimum is less than or equal to the infimum. (Contributed by Jim Kingdon, 27-Sep-2024.) |
Ref | Expression |
---|---|
infregelbex.ex | |
infregelbex.ss | |
infregelbex.b |
Ref | Expression |
---|---|
infregelbex | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infregelbex.b | . . . . . 6 | |
2 | 1 | ad2antrr 480 | . . . . 5 inf |
3 | lttri3 7960 | . . . . . . . 8 | |
4 | 3 | adantl 275 | . . . . . . 7 |
5 | infregelbex.ex | . . . . . . 7 | |
6 | 4, 5 | infclti 6970 | . . . . . 6 inf |
7 | 6 | ad2antrr 480 | . . . . 5 inf inf |
8 | infregelbex.ss | . . . . . . 7 | |
9 | 8 | sselda 3128 | . . . . . 6 |
10 | 9 | adantlr 469 | . . . . 5 inf |
11 | simplr 520 | . . . . 5 inf inf | |
12 | 6 | adantr 274 | . . . . . . 7 inf |
13 | 4, 5 | inflbti 6971 | . . . . . . . 8 inf |
14 | 13 | imp 123 | . . . . . . 7 inf |
15 | 12, 9, 14 | nltled 8001 | . . . . . 6 inf |
16 | 15 | adantlr 469 | . . . . 5 inf inf |
17 | 2, 7, 10, 11, 16 | letrd 8004 | . . . 4 inf |
18 | 17 | ralrimiva 2530 | . . 3 inf |
19 | breq2 3971 | . . . 4 | |
20 | 19 | cbvralv 2680 | . . 3 |
21 | 18, 20 | sylib 121 | . 2 inf |
22 | 1 | adantr 274 | . . 3 |
23 | 6 | adantr 274 | . . 3 inf |
24 | simpl 108 | . . . 4 | |
25 | simpr 109 | . . . . . . . 8 | |
26 | breq2 3971 | . . . . . . . . 9 | |
27 | 26 | cbvralv 2680 | . . . . . . . 8 |
28 | 25, 27 | sylib 121 | . . . . . . 7 |
29 | 1 | ad2antrr 480 | . . . . . . . . 9 |
30 | 8 | ad2antrr 480 | . . . . . . . . . 10 |
31 | simpr 109 | . . . . . . . . . 10 | |
32 | 30, 31 | sseldd 3129 | . . . . . . . . 9 |
33 | 29, 32 | lenltd 7998 | . . . . . . . 8 |
34 | 33 | ralbidva 2453 | . . . . . . 7 |
35 | 28, 34 | mpbid 146 | . . . . . 6 |
36 | breq1 3970 | . . . . . . . 8 | |
37 | 36 | notbid 657 | . . . . . . 7 |
38 | 37 | cbvralv 2680 | . . . . . 6 |
39 | 35, 38 | sylib 121 | . . . . 5 |
40 | 22, 39 | jca 304 | . . . 4 |
41 | 4, 5 | infnlbti 6973 | . . . 4 inf |
42 | 24, 40, 41 | sylc 62 | . . 3 inf |
43 | 22, 23, 42 | nltled 8001 | . 2 inf |
44 | 21, 43 | impbida 586 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wcel 2128 wral 2435 wrex 2436 wss 3102 class class class wbr 3967 infcinf 6930 cr 7734 clt 7915 cle 7916 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-cnex 7826 ax-resscn 7827 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-apti 7850 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4029 df-xp 4595 df-cnv 4597 df-iota 5138 df-riota 5783 df-sup 6931 df-inf 6932 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 |
This theorem is referenced by: nninfdclemp1 12277 |
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