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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 9965 | . 2 ⊢ 0 ∈ ℝ* | |
2 | pnfge 11840 | . 2 ⊢ (0 ∈ ℝ* → 0 ≤ +∞) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 0 ≤ +∞ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 class class class wbr 4583 0cc0 9815 +∞cpnf 9950 ℝ*cxr 9952 ≤ cle 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-iota 5768 df-fv 5812 df-ov 6552 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 |
This theorem is referenced by: xnn0ge0 11843 nn0pnfge0OLD 11844 xsubge0 11963 xadddi2 11999 xnn0xrge0 12196 pcge0 15404 leordtval2 20826 iccpnfcnv 22551 taylfval 23917 elxrge02 28971 xrge0adddir 29023 xrge0iifcnv 29307 lmxrge0 29326 esumpinfval 29462 hashf2 29473 esumcvg 29475 pnfel0pnf 38601 |
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