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Mirrors > Home > ILE Home > Th. List > infssuzledc | GIF version |
Description: The infimum of a subset of an upper set of integers is less than or equal to all members of the subset. (Contributed by Jim Kingdon, 13-Jan-2022.) |
Ref | Expression |
---|---|
infssuzledc.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
infssuzledc.s | ⊢ 𝑆 = {𝑛 ∈ (ℤ≥‘𝑀) ∣ 𝜓} |
infssuzledc.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
infssuzledc.dc | ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓) |
Ref | Expression |
---|---|
infssuzledc | ⊢ (𝜑 → inf(𝑆, ℝ, < ) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lttri3 7812 | . . . 4 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 = 𝑏 ↔ (¬ 𝑎 < 𝑏 ∧ ¬ 𝑏 < 𝑎))) | |
2 | 1 | adantl 275 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ)) → (𝑎 = 𝑏 ↔ (¬ 𝑎 < 𝑏 ∧ ¬ 𝑏 < 𝑎))) |
3 | infssuzledc.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
4 | infssuzledc.s | . . . 4 ⊢ 𝑆 = {𝑛 ∈ (ℤ≥‘𝑀) ∣ 𝜓} | |
5 | infssuzledc.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
6 | infssuzledc.dc | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓) | |
7 | 3, 4, 5, 6 | infssuzex 11569 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝑆 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝑆 𝑧 < 𝑦))) |
8 | 2, 7 | infclti 6878 | . 2 ⊢ (𝜑 → inf(𝑆, ℝ, < ) ∈ ℝ) |
9 | elrabi 2810 | . . . 4 ⊢ (𝐴 ∈ {𝑛 ∈ (ℤ≥‘𝑀) ∣ 𝜓} → 𝐴 ∈ (ℤ≥‘𝑀)) | |
10 | 9, 4 | eleq2s 2212 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ (ℤ≥‘𝑀)) |
11 | eluzelre 9304 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘𝑀) → 𝐴 ∈ ℝ) | |
12 | 5, 10, 11 | 3syl 17 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
13 | 2, 7 | inflbti 6879 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑆 → ¬ 𝐴 < inf(𝑆, ℝ, < ))) |
14 | 5, 13 | mpd 13 | . 2 ⊢ (𝜑 → ¬ 𝐴 < inf(𝑆, ℝ, < )) |
15 | 8, 12, 14 | nltled 7851 | 1 ⊢ (𝜑 → inf(𝑆, ℝ, < ) ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 DECID wdc 804 = wceq 1316 ∈ wcel 1465 {crab 2397 class class class wbr 3899 ‘cfv 5093 (class class class)co 5742 infcinf 6838 ℝcr 7587 < clt 7768 ≤ cle 7769 ℤcz 9022 ℤ≥cuz 9294 ...cfz 9758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-sup 6839 df-inf 6840 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 df-uz 9295 df-fz 9759 df-fzo 9888 |
This theorem is referenced by: lcmledvds 11678 |
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