![]() |
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 10632 | . 2 ⊢ 0 ∈ ℝ | |
2 | 0le0 11726 | . 2 ⊢ 0 ≤ 0 | |
3 | elrege0 12832 | . 2 ⊢ (0 ∈ (0[,)+∞) ↔ (0 ∈ ℝ ∧ 0 ≤ 0)) | |
4 | 1, 2, 3 | mpbir2an 710 | 1 ⊢ 0 ∈ (0[,)+∞) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 +∞cpnf 10661 ≤ cle 10665 [,)cico 12728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-addrcl 10587 ax-rnegex 10597 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ico 12732 |
This theorem is referenced by: fsumge0 15142 rege0subm 20147 rge0srg 20162 itg2cnlem1 24365 ibladdlem 24423 itgaddlem1 24426 iblabslem 24431 iblabs 24432 iblmulc2 24434 itgmulc2lem1 24435 bddmulibl 24442 itggt0 24447 itgcn 24448 cxpcn3 25337 rlimcnp3 25553 efrlim 25555 fsumrp0cl 30729 xrge0slmod 30968 esumpfinvallem 31443 ibladdnclem 35113 itgaddnclem1 35115 iblabsnclem 35120 iblabsnc 35121 iblmulc2nc 35122 itgmulc2nclem1 35123 itggt0cn 35127 ftc1anclem8 35137 sge0z 43014 sge0tsms 43019 hoidmvcl 43221 dig0 45020 |
Copyright terms: Public domain | W3C validator |