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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 11850 | . . . 4 ⊢ < Or ℝ* | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → < Or ℝ*) |
3 | pnfxr 9971 | . . . 4 ⊢ +∞ ∈ ℝ* | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → +∞ ∈ ℝ*) |
5 | noel 3878 | . . . . 5 ⊢ ¬ 𝑦 ∈ ∅ | |
6 | 5 | pm2.21i 115 | . . . 4 ⊢ (𝑦 ∈ ∅ → ¬ 𝑦 < +∞) |
7 | 6 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ∅) → ¬ 𝑦 < +∞) |
8 | pnfnlt 11838 | . . . . . 6 ⊢ (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦) | |
9 | 8 | pm2.21d 117 | . . . . 5 ⊢ (𝑦 ∈ ℝ* → (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)) |
10 | 9 | imp 444 | . . . 4 ⊢ ((𝑦 ∈ ℝ* ∧ +∞ < 𝑦) → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) |
11 | 10 | adantl 481 | . . 3 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ* ∧ +∞ < 𝑦)) → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) |
12 | 2, 4, 7, 11 | eqinfd 8274 | . 2 ⊢ (⊤ → inf(∅, ℝ*, < ) = +∞) |
13 | 12 | trud 1484 | 1 ⊢ inf(∅, ℝ*, < ) = +∞ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ∃wrex 2897 ∅c0 3874 class class class wbr 4583 Or wor 4958 infcinf 8230 +∞cpnf 9950 ℝ*cxr 9952 < clt 9953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 |
This theorem is referenced by: ramcl2lem 15551 infleinf 38529 |
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