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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 13069 | . . . 4 ⊢ < Or ℝ* | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → < Or ℝ*) |
3 | pnfxr 11217 | . . . 4 ⊢ +∞ ∈ ℝ* | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → +∞ ∈ ℝ*) |
5 | noel 4294 | . . . . 5 ⊢ ¬ 𝑦 ∈ ∅ | |
6 | 5 | pm2.21i 119 | . . . 4 ⊢ (𝑦 ∈ ∅ → ¬ 𝑦 < +∞) |
7 | 6 | adantl 483 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ∅) → ¬ 𝑦 < +∞) |
8 | pnfnlt 13057 | . . . . . 6 ⊢ (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦) | |
9 | 8 | pm2.21d 121 | . . . . 5 ⊢ (𝑦 ∈ ℝ* → (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)) |
10 | 9 | imp 408 | . . . 4 ⊢ ((𝑦 ∈ ℝ* ∧ +∞ < 𝑦) → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) |
11 | 10 | adantl 483 | . . 3 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ* ∧ +∞ < 𝑦)) → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) |
12 | 2, 4, 7, 11 | eqinfd 9429 | . 2 ⊢ (⊤ → inf(∅, ℝ*, < ) = +∞) |
13 | 12 | mptru 1549 | 1 ⊢ inf(∅, ℝ*, < ) = +∞ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ∃wrex 3070 ∅c0 4286 class class class wbr 5109 Or wor 5548 infcinf 9385 +∞cpnf 11194 ℝ*cxr 11196 < clt 11197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-pre-lttri 11133 ax-pre-lttrn 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 |
This theorem is referenced by: ramcl2lem 16889 infleinf 43697 infxrpnf 43771 supxrltinfxr 43774 supminfxr 43789 |
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