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Mirrors > Home > MPE Home > Th. List > pnfge | Structured version Visualization version Unicode version |
Description: Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.) |
Ref | Expression |
---|---|
pnfge |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfnlt 11962 | . 2 | |
2 | pnfxr 10092 | . . 3 | |
3 | xrlenlt 10103 | . . 3 | |
4 | 2, 3 | mpan2 707 | . 2 |
5 | 1, 4 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wcel 1990 class class class wbr 4653 cpnf 10071 cxr 10073 clt 10074 cle 10075 |
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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-rab 2921 df-v 3202 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-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 |
This theorem is referenced by: xnn0n0n1ge2b 11965 0lepnf 11966 nltpnft 11995 xrre2 12001 xleadd1a 12083 xlt2add 12090 xsubge0 12091 xlesubadd 12093 xlemul1a 12118 elicore 12226 elico2 12237 iccmax 12249 elxrge0 12281 nfile 13150 hashdom 13168 hashdomi 13169 hashge1 13178 hashss 13197 hashge2el2difr 13263 pcdvdsb 15573 pc2dvds 15583 pcaddlem 15592 xrsdsreclblem 19792 leordtvallem1 21014 lecldbas 21023 isxmet2d 22132 blssec 22240 nmoix 22533 nmoleub 22535 xrtgioo 22609 xrge0tsms 22637 metdstri 22654 nmoleub2lem 22914 ovolf 23250 ovollb2 23257 ovoliun 23273 ovolre 23293 voliunlem3 23320 volsup 23324 icombl 23332 ioombl 23333 ismbfd 23407 itg2seq 23509 dvfsumrlim 23794 dvfsumrlim2 23795 radcnvcl 24171 xrlimcnp 24695 logfacbnd3 24948 log2sumbnd 25233 tgldimor 25397 xrdifh 29542 xrge0tsmsd 29785 unitssxrge0 29946 tpr2rico 29958 lmxrge0 29998 esumle 30120 esumlef 30124 esumpinfval 30135 esumpinfsum 30139 esumcvgsum 30150 ddemeas 30299 sxbrsigalem2 30348 omssubadd 30362 carsgclctunlem3 30382 signsply0 30628 ismblfin 33450 xrgepnfd 39547 supxrgelem 39553 supxrge 39554 infrpge 39567 xrlexaddrp 39568 infleinflem1 39586 infleinf 39588 infxrpnf 39674 pnfged 39704 ge0xrre 39758 iblsplit 40182 ismbl3 40203 ovolsplit 40205 sge0cl 40598 sge0less 40609 sge0pr 40611 sge0le 40624 sge0split 40626 carageniuncl 40737 ovnsubaddlem1 40784 hspmbl 40843 pimiooltgt 40921 pgrpgt2nabl 42147 |
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