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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 7468 | . . . 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 10725 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝑆 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝑆 𝑧 < 𝑦))) |
8 | 2, 7 | infclti 6625 | . 2 ⊢ (𝜑 → inf(𝑆, ℝ, < ) ∈ ℝ) |
9 | elrabi 2756 | . . . 4 ⊢ (𝐴 ∈ {𝑛 ∈ (ℤ≥‘𝑀) ∣ 𝜓} → 𝐴 ∈ (ℤ≥‘𝑀)) | |
10 | 9, 4 | eleq2s 2177 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ (ℤ≥‘𝑀)) |
11 | eluzelre 8924 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘𝑀) → 𝐴 ∈ ℝ) | |
12 | 5, 10, 11 | 3syl 17 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
13 | 2, 7 | inflbti 6626 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑆 → ¬ 𝐴 < inf(𝑆, ℝ, < ))) |
14 | 5, 13 | mpd 13 | . 2 ⊢ (𝜑 → ¬ 𝐴 < inf(𝑆, ℝ, < )) |
15 | 8, 12, 14 | nltled 7507 | 1 ⊢ (𝜑 → inf(𝑆, ℝ, < ) ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 DECID wdc 776 = wceq 1285 ∈ wcel 1434 {crab 2357 class class class wbr 3811 ‘cfv 4969 (class class class)co 5591 infcinf 6585 ℝcr 7252 < clt 7425 ≤ cle 7426 ℤcz 8646 ℤ≥cuz 8914 ...cfz 9319 |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-addcom 7348 ax-addass 7350 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-0id 7356 ax-rnegex 7357 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-apti 7363 ax-pre-ltadd 7364 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-id 4084 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-1st 5846 df-2nd 5847 df-sup 6586 df-inf 6587 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-inn 8317 df-n0 8566 df-z 8647 df-uz 8915 df-fz 9320 df-fzo 9444 |
This theorem is referenced by: lcmledvds 10832 |
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