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Mirrors > Home > MPE Home > Th. List > ge0nemnf | Structured version Visualization version Unicode version |
Description: A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
ge0nemnf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ge0gtmnf 12003 | . 2 | |
2 | ngtmnft 11997 | . . . 4 | |
3 | 2 | adantr 481 | . . 3 |
4 | 3 | necon2abid 2836 | . 2 |
5 | 1, 4 | mpbid 222 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wne 2794 class class class wbr 4653 cc0 9936 cmnf 10072 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 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
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-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-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-ov 6653 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 |
This theorem is referenced by: xlesubadd 12093 xadddi2 12127 xrge0subm 19787 isxmet2d 22132 xmetrtri 22160 imasdsf1olem 22178 xblpnfps 22200 xblpnf 22201 xblss2ps 22206 xblss2 22207 nmoix 22533 nmoleub 22535 blcvx 22601 xrge0gsumle 22636 xrge0tsms 22637 metdstri 22654 metdscnlem 22658 nmoleub2lem 22914 xrge0addass 29690 xrge0tsmsd 29785 ge0nemnf2 39755 |
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