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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 10676 | . 2 ⊢ 0 ∈ ℝ* | |
2 | 0le0 11726 | . 2 ⊢ 0 ≤ 0 | |
3 | elxrge0 12833 | . 2 ⊢ (0 ∈ (0[,]+∞) ↔ (0 ∈ ℝ* ∧ 0 ≤ 0)) | |
4 | 1, 2, 3 | mpbir2an 707 | 1 ⊢ 0 ∈ (0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 0cc0 10525 +∞cpnf 10660 ℝ*cxr 10662 ≤ cle 10664 [,]cicc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-addrcl 10586 ax-rnegex 10596 ax-cnre 10598 ax-pre-lttri 10599 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-icc 12733 |
This theorem is referenced by: xrge0subm 20514 itg2const2 24269 itg2splitlem 24276 itg2split 24277 itg2gt0 24288 itg2cnlem2 24290 itg2cn 24291 iblss 24332 itgle 24337 itgeqa 24341 ibladdlem 24347 iblabs 24356 iblabsr 24357 iblmulc2 24358 bddmulibl 24366 xrge0infss 30410 xrge00 30600 unitssxrge0 31042 xrge0mulc1cn 31083 esum0 31207 esumpad 31213 esumpad2 31214 esumrnmpt2 31226 esumpinfval 31231 esummulc1 31239 ddemeas 31394 oms0 31454 itg2gt0cn 34828 ibladdnclem 34829 iblabsnc 34837 iblmulc2nc 34838 bddiblnc 34843 ftc1anclem7 34854 ftc1anclem8 34855 ftc1anc 34856 iblsplit 42127 gsumge0cl 42530 sge0cl 42540 sge0ss 42571 0ome 42688 ovnf 42722 |
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