Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 0e0icopnf | Structured version Visualization version GIF version |
Description: 0 is a member of (0[,)+∞). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
0e0icopnf | ⊢ 0 ∈ (0[,)+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10977 | . 2 ⊢ 0 ∈ ℝ | |
2 | 0le0 12074 | . 2 ⊢ 0 ≤ 0 | |
3 | elrege0 13186 | . 2 ⊢ (0 ∈ (0[,)+∞) ↔ (0 ∈ ℝ ∧ 0 ≤ 0)) | |
4 | 1, 2, 3 | mpbir2an 708 | 1 ⊢ 0 ∈ (0[,)+∞) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 class class class wbr 5074 (class class class)co 7275 ℝcr 10870 0cc0 10871 +∞cpnf 11006 ≤ cle 11010 [,)cico 13081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-addrcl 10932 ax-rnegex 10942 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-ico 13085 |
This theorem is referenced by: fsumge0 15507 rege0subm 20654 rge0srg 20669 itg2cnlem1 24926 ibladdlem 24984 itgaddlem1 24987 iblabslem 24992 iblabs 24993 iblmulc2 24995 itgmulc2lem1 24996 bddmulibl 25003 itggt0 25008 itgcn 25009 cxpcn3 25901 rlimcnp3 26117 efrlim 26119 fsumrp0cl 31304 xrge0slmod 31548 esumpfinvallem 32042 ibladdnclem 35833 itgaddnclem1 35835 iblabsnclem 35840 iblabsnc 35841 iblmulc2nc 35842 itgmulc2nclem1 35843 itggt0cn 35847 ftc1anclem8 35857 sge0z 43913 sge0tsms 43918 hoidmvcl 44120 dig0 45952 |
Copyright terms: Public domain | W3C validator |