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| 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 13184 | . . . 4 ⊢ < Or ℝ* | |
| 2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → < Or ℝ*) | 
| 3 | pnfxr 11316 | . . . 4 ⊢ +∞ ∈ ℝ* | |
| 4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → +∞ ∈ ℝ*) | 
| 5 | noel 4337 | . . . . 5 ⊢ ¬ 𝑦 ∈ ∅ | |
| 6 | 5 | pm2.21i 119 | . . . 4 ⊢ (𝑦 ∈ ∅ → ¬ 𝑦 < +∞) | 
| 7 | 6 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ∅) → ¬ 𝑦 < +∞) | 
| 8 | pnfnlt 13171 | . . . . . 6 ⊢ (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦) | |
| 9 | 8 | pm2.21d 121 | . . . . 5 ⊢ (𝑦 ∈ ℝ* → (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)) | 
| 10 | 9 | imp 406 | . . . 4 ⊢ ((𝑦 ∈ ℝ* ∧ +∞ < 𝑦) → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) | 
| 11 | 10 | adantl 481 | . . 3 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ* ∧ +∞ < 𝑦)) → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) | 
| 12 | 2, 4, 7, 11 | eqinfd 9526 | . 2 ⊢ (⊤ → inf(∅, ℝ*, < ) = +∞) | 
| 13 | 12 | mptru 1546 | 1 ⊢ inf(∅, ℝ*, < ) = +∞ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1539 ⊤wtru 1540 ∈ wcel 2107 ∃wrex 3069 ∅c0 4332 class class class wbr 5142 Or wor 5590 infcinf 9482 +∞cpnf 11293 ℝ*cxr 11295 < clt 11296 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-pre-lttri 11230 ax-pre-lttrn 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 | 
| This theorem is referenced by: ramcl2lem 17048 infleinf 45388 infxrpnf 45462 supxrltinfxr 45465 supminfxr 45480 | 
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