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| Mirrors > Home > MPE Home > Th. List > 0lepnf | Structured version Visualization version GIF version | ||
| Description: 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.) |
| Ref | Expression |
|---|---|
| 0lepnf | ⊢ 0 ≤ +∞ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0xr 11308 | . 2 ⊢ 0 ∈ ℝ* | |
| 2 | pnfge 13172 | . 2 ⊢ (0 ∈ ℝ* → 0 ≤ +∞) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ 0 ≤ +∞ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 class class class wbr 5143 0cc0 11155 +∞cpnf 11292 ℝ*cxr 11294 ≤ cle 11296 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 |
| This theorem is referenced by: xnn0ge0 13176 xsubge0 13303 xadddi2 13339 xnn0xrge0 13546 pcge0 16900 leordtval2 23220 iccpnfcnv 24975 taylfval 26400 elxrge02 32914 xrge0adddir 33023 xrge0iifcnv 33932 lmxrge0 33951 esumpinfval 34074 hashf2 34085 esumcvg 34087 aks4d1p1p6 42074 pnfel0pnf 45541 |
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