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Mirrors > Home > MPE Home > Th. List > xrinf0 | Structured version Visualization version GIF version |
Description: The infimum of the empty set under the extended reals is positive infinity. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by AV, 5-Sep-2020.) |
Ref | Expression |
---|---|
xrinf0 | ⊢ inf(∅, ℝ*, < ) = +∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltso 12524 | . . . 4 ⊢ < Or ℝ* | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → < Or ℝ*) |
3 | pnfxr 10684 | . . . 4 ⊢ +∞ ∈ ℝ* | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → +∞ ∈ ℝ*) |
5 | noel 4295 | . . . . 5 ⊢ ¬ 𝑦 ∈ ∅ | |
6 | 5 | pm2.21i 119 | . . . 4 ⊢ (𝑦 ∈ ∅ → ¬ 𝑦 < +∞) |
7 | 6 | adantl 482 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ∅) → ¬ 𝑦 < +∞) |
8 | pnfnlt 12513 | . . . . . 6 ⊢ (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦) | |
9 | 8 | pm2.21d 121 | . . . . 5 ⊢ (𝑦 ∈ ℝ* → (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)) |
10 | 9 | imp 407 | . . . 4 ⊢ ((𝑦 ∈ ℝ* ∧ +∞ < 𝑦) → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) |
11 | 10 | adantl 482 | . . 3 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ* ∧ +∞ < 𝑦)) → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) |
12 | 2, 4, 7, 11 | eqinfd 8938 | . 2 ⊢ (⊤ → inf(∅, ℝ*, < ) = +∞) |
13 | 12 | mptru 1535 | 1 ⊢ inf(∅, ℝ*, < ) = +∞ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 ∃wrex 3139 ∅c0 4290 class class class wbr 5058 Or wor 5467 infcinf 8894 +∞cpnf 10661 ℝ*cxr 10663 < clt 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-sup 8895 df-inf 8896 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 |
This theorem is referenced by: ramcl2lem 16335 infleinf 41520 infxrpnf 41601 supxrltinfxr 41604 supminfxr 41620 |
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