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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 13155 | . . . 4 ⊢ < Or ℝ* | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → < Or ℝ*) |
3 | pnfxr 11300 | . . . 4 ⊢ +∞ ∈ ℝ* | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → +∞ ∈ ℝ*) |
5 | noel 4330 | . . . . 5 ⊢ ¬ 𝑦 ∈ ∅ | |
6 | 5 | pm2.21i 119 | . . . 4 ⊢ (𝑦 ∈ ∅ → ¬ 𝑦 < +∞) |
7 | 6 | adantl 480 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ∅) → ¬ 𝑦 < +∞) |
8 | pnfnlt 13143 | . . . . . 6 ⊢ (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦) | |
9 | 8 | pm2.21d 121 | . . . . 5 ⊢ (𝑦 ∈ ℝ* → (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)) |
10 | 9 | imp 405 | . . . 4 ⊢ ((𝑦 ∈ ℝ* ∧ +∞ < 𝑦) → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) |
11 | 10 | adantl 480 | . . 3 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ* ∧ +∞ < 𝑦)) → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) |
12 | 2, 4, 7, 11 | eqinfd 9510 | . 2 ⊢ (⊤ → inf(∅, ℝ*, < ) = +∞) |
13 | 12 | mptru 1540 | 1 ⊢ inf(∅, ℝ*, < ) = +∞ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 394 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ∃wrex 3059 ∅c0 4322 class class class wbr 5149 Or wor 5589 infcinf 9466 +∞cpnf 11277 ℝ*cxr 11279 < clt 11280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-pre-lttri 11214 ax-pre-lttrn 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 |
This theorem is referenced by: ramcl2lem 16981 infleinf 44892 infxrpnf 44966 supxrltinfxr 44969 supminfxr 44984 |
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