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Mirrors > Home > MPE Home > Th. List > Mathboxes > pnfel0pnf | Structured version Visualization version GIF version |
Description: +∞ is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
pnfel0pnf | ⊢ +∞ ∈ (0[,]+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 10863 | . 2 ⊢ 0 ∈ ℝ* | |
2 | pnfxr 10870 | . 2 ⊢ +∞ ∈ ℝ* | |
3 | 0lepnf 12707 | . 2 ⊢ 0 ≤ +∞ | |
4 | ubicc2 13036 | . 2 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞)) | |
5 | 1, 2, 3, 4 | mp3an 1463 | 1 ⊢ +∞ ∈ (0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 class class class wbr 5043 (class class class)co 7202 0cc0 10712 +∞cpnf 10847 ℝ*cxr 10849 ≤ cle 10851 [,]cicc 12921 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-addrcl 10773 ax-rnegex 10783 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-po 5457 df-so 5458 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-icc 12925 |
This theorem is referenced by: sge0cl 43548 |
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