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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 7191 | . . . 4 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 = 𝑏 ↔ (¬ 𝑎 < 𝑏 ∧ ¬ 𝑏 < 𝑎))) | |
2 | 1 | adantl 271 | . . 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 10345 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝑆 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝑆 𝑧 < 𝑦))) |
8 | 2, 7 | infclti 6436 | . 2 ⊢ (𝜑 → inf(𝑆, ℝ, < ) ∈ ℝ) |
9 | elrabi 2746 | . . . 4 ⊢ (𝐴 ∈ {𝑛 ∈ (ℤ≥‘𝑀) ∣ 𝜓} → 𝐴 ∈ (ℤ≥‘𝑀)) | |
10 | 9, 4 | eleq2s 2173 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ (ℤ≥‘𝑀)) |
11 | eluzelre 8629 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘𝑀) → 𝐴 ∈ ℝ) | |
12 | 5, 10, 11 | 3syl 17 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
13 | 2, 7 | inflbti 6437 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑆 → ¬ 𝐴 < inf(𝑆, ℝ, < ))) |
14 | 5, 13 | mpd 13 | . 2 ⊢ (𝜑 → ¬ 𝐴 < inf(𝑆, ℝ, < )) |
15 | 8, 12, 14 | nltled 7230 | 1 ⊢ (𝜑 → inf(𝑆, ℝ, < ) ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 DECID wdc 775 = wceq 1284 ∈ wcel 1433 {crab 2352 class class class wbr 3785 ‘cfv 4922 (class class class)co 5532 infcinf 6396 ℝcr 6980 < clt 7153 ≤ cle 7154 ℤcz 8351 ℤ≥cuz 8619 ...cfz 9029 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-sup 6397 df-inf 6398 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-fz 9030 df-fzo 9153 |
This theorem is referenced by: lcmledvds 10452 |
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