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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 10845 | . 2 ⊢ 0 ∈ ℝ* | |
2 | pnfge 12687 | . 2 ⊢ (0 ∈ ℝ* → 0 ≤ +∞) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 0 ≤ +∞ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2112 class class class wbr 5039 0cc0 10694 +∞cpnf 10829 ℝ*cxr 10831 ≤ cle 10833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-addrcl 10755 ax-rnegex 10765 ax-cnre 10767 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-xp 5542 df-cnv 5544 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 |
This theorem is referenced by: xnn0ge0 12690 xsubge0 12816 xadddi2 12852 xnn0xrge0 13059 pcge0 16378 leordtval2 22063 iccpnfcnv 23795 taylfval 25205 elxrge02 30880 xrge0adddir 30974 xrge0iifcnv 31551 lmxrge0 31570 esumpinfval 31707 hashf2 31718 esumcvg 31720 aks4d1p1p6 39763 pnfel0pnf 42682 |
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