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Mirrors > Home > MPE Home > Th. List > 0e0iccpnf | Structured version Visualization version GIF version |
Description: 0 is a member of (0[,]+∞). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
0e0iccpnf | ⊢ 0 ∈ (0[,]+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 10278 | . 2 ⊢ 0 ∈ ℝ* | |
2 | 0le0 11302 | . 2 ⊢ 0 ≤ 0 | |
3 | elxrge0 12474 | . 2 ⊢ (0 ∈ (0[,]+∞) ↔ (0 ∈ ℝ* ∧ 0 ≤ 0)) | |
4 | 1, 2, 3 | mpbir2an 993 | 1 ⊢ 0 ∈ (0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2139 class class class wbr 4804 (class class class)co 6813 0cc0 10128 +∞cpnf 10263 ℝ*cxr 10265 ≤ cle 10267 [,]cicc 12371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-i2m1 10196 ax-1ne0 10197 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-icc 12375 |
This theorem is referenced by: xrge0subm 19989 itg2const2 23707 itg2splitlem 23714 itg2split 23715 itg2gt0 23726 itg2cnlem2 23728 itg2cn 23729 iblss 23770 itgle 23775 itgeqa 23779 ibladdlem 23785 iblabs 23794 iblabsr 23795 iblmulc2 23796 bddmulibl 23804 xrge0infss 29834 xrge00 29995 unitssxrge0 30255 xrge0mulc1cn 30296 esum0 30420 esumpad 30426 esumpad2 30427 esumrnmpt2 30439 esumpinfval 30444 esummulc1 30452 ddemeas 30608 oms0 30668 itg2gt0cn 33778 ibladdnclem 33779 iblabsnc 33787 iblmulc2nc 33788 bddiblnc 33793 ftc1anclem7 33804 ftc1anclem8 33805 ftc1anc 33806 iblsplit 40685 gsumge0cl 41091 sge0cl 41101 sge0ss 41132 0ome 41249 ovnf 41283 |
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