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Mirrors > Home > MPE Home > Th. List > 0e0iccpnf | Structured version Visualization version GIF version |
Description: 0 is a member of (0[,]+∞) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
0e0iccpnf | ⊢ 0 ∈ (0[,]+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 10086 | . 2 ⊢ 0 ∈ ℝ* | |
2 | 0le0 11110 | . 2 ⊢ 0 ≤ 0 | |
3 | elxrge0 12281 | . 2 ⊢ (0 ∈ (0[,]+∞) ↔ (0 ∈ ℝ* ∧ 0 ≤ 0)) | |
4 | 1, 2, 3 | mpbir2an 955 | 1 ⊢ 0 ∈ (0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 0cc0 9936 +∞cpnf 10071 ℝ*cxr 10073 ≤ cle 10075 [,]cicc 12178 |
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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-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-oprab 6654 df-mpt2 6655 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 df-icc 12182 |
This theorem is referenced by: xrge0subm 19787 itg2const2 23508 itg2splitlem 23515 itg2split 23516 itg2gt0 23527 itg2cnlem2 23529 itg2cn 23530 iblss 23571 itgle 23576 itgeqa 23580 ibladdlem 23586 iblabs 23595 iblabsr 23596 iblmulc2 23597 bddmulibl 23605 xrge0infss 29525 xrge00 29686 unitssxrge0 29946 xrge0mulc1cn 29987 esum0 30111 esumpad 30117 esumpad2 30118 esumrnmpt2 30130 esumpinfval 30135 esummulc1 30143 ddemeas 30299 oms0 30359 itg2gt0cn 33465 ibladdnclem 33466 iblabsnc 33474 iblmulc2nc 33475 bddiblnc 33480 ftc1anclem7 33491 ftc1anclem8 33492 ftc1anc 33493 iblsplit 40182 gsumge0cl 40588 sge0cl 40598 sge0ss 40629 0ome 40743 ovnf 40777 |
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