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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 10734 | . 2 ⊢ 0 ∈ ℝ | |
2 | 0le0 11830 | . 2 ⊢ 0 ≤ 0 | |
3 | elrege0 12941 | . 2 ⊢ (0 ∈ (0[,)+∞) ↔ (0 ∈ ℝ ∧ 0 ≤ 0)) | |
4 | 1, 2, 3 | mpbir2an 711 | 1 ⊢ 0 ∈ (0[,)+∞) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 class class class wbr 5040 (class class class)co 7183 ℝcr 10627 0cc0 10628 +∞cpnf 10763 ≤ cle 10767 [,)cico 12836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-addrcl 10689 ax-rnegex 10699 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7186 df-oprab 7187 df-mpo 7188 df-er 8333 df-en 8569 df-dom 8570 df-sdom 8571 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-ico 12840 |
This theorem is referenced by: fsumge0 15256 rege0subm 20286 rge0srg 20301 itg2cnlem1 24527 ibladdlem 24585 itgaddlem1 24588 iblabslem 24593 iblabs 24594 iblmulc2 24596 itgmulc2lem1 24597 bddmulibl 24604 itggt0 24609 itgcn 24610 cxpcn3 25502 rlimcnp3 25718 efrlim 25720 fsumrp0cl 30894 xrge0slmod 31133 esumpfinvallem 31625 ibladdnclem 35489 itgaddnclem1 35491 iblabsnclem 35496 iblabsnc 35497 iblmulc2nc 35498 itgmulc2nclem1 35499 itggt0cn 35503 ftc1anclem8 35513 sge0z 43496 sge0tsms 43501 hoidmvcl 43703 dig0 45534 |
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