Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ressiooinf | Structured version Visualization version Unicode version |
Description: If the infimum does not belong to a set of reals, the set is a subset of the unbounded above, left-open interval, with lower bound equal to the infimum. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
ressiooinf.a | |
ressiooinf.s | inf |
ressiooinf.n | |
ressiooinf.i |
Ref | Expression |
---|---|
ressiooinf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressiooinf.s | . . . . . 6 inf | |
2 | ressiooinf.a | . . . . . . . . 9 | |
3 | ressxr 10083 | . . . . . . . . . 10 | |
4 | 3 | a1i 11 | . . . . . . . . 9 |
5 | 2, 4 | sstrd 3613 | . . . . . . . 8 |
6 | 5 | adantr 481 | . . . . . . 7 |
7 | 6 | infxrcld 39612 | . . . . . 6 inf |
8 | 1, 7 | syl5eqel 2705 | . . . . 5 |
9 | pnfxr 10092 | . . . . . 6 | |
10 | 9 | a1i 11 | . . . . 5 |
11 | 2 | adantr 481 | . . . . . 6 |
12 | simpr 477 | . . . . . 6 | |
13 | 11, 12 | sseldd 3604 | . . . . 5 |
14 | 5 | sselda 3603 | . . . . . 6 |
15 | infxrlb 12164 | . . . . . . . 8 inf | |
16 | 6, 12, 15 | syl2anc 693 | . . . . . . 7 inf |
17 | 1, 16 | syl5eqbr 4688 | . . . . . 6 |
18 | id 22 | . . . . . . . . . . . . 13 | |
19 | 18 | eqcomd 2628 | . . . . . . . . . . . 12 |
20 | 19 | adantl 482 | . . . . . . . . . . 11 |
21 | simpl 473 | . . . . . . . . . . 11 | |
22 | 20, 21 | eqeltrd 2701 | . . . . . . . . . 10 |
23 | 22 | adantll 750 | . . . . . . . . 9 |
24 | ressiooinf.n | . . . . . . . . . 10 | |
25 | 24 | ad2antrr 762 | . . . . . . . . 9 |
26 | 23, 25 | pm2.65da 600 | . . . . . . . 8 |
27 | 26 | neqned 2801 | . . . . . . 7 |
28 | 27 | necomd 2849 | . . . . . 6 |
29 | 8, 14, 17, 28 | xrleneltd 39539 | . . . . 5 |
30 | 13 | ltpnfd 11955 | . . . . 5 |
31 | 8, 10, 13, 29, 30 | eliood 39720 | . . . 4 |
32 | ressiooinf.i | . . . 4 | |
33 | 31, 32 | syl6eleqr 2712 | . . 3 |
34 | 33 | ralrimiva 2966 | . 2 |
35 | dfss3 3592 | . 2 | |
36 | 34, 35 | sylibr 224 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 wss 3574 class class class wbr 4653 (class class class)co 6650 infcinf 8347 cr 9935 cpnf 10071 cxr 10073 clt 10074 cle 10075 cioo 12175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-ioo 12179 |
This theorem is referenced by: (None) |
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