![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fiminre2 | Structured version Visualization version GIF version |
Description: A nonempty finite set of real numbers is bounded below. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
fiminre2 | ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 10479 | . . . 4 ⊢ (𝐴 = ∅ → 0 ∈ ℝ) | |
2 | rzal 4361 | . . . 4 ⊢ (𝐴 = ∅ → ∀𝑦 ∈ 𝐴 0 ≤ 𝑦) | |
3 | breq1 4959 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦)) | |
4 | 3 | ralbidv 3162 | . . . . 5 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝐴 0 ≤ 𝑦)) |
5 | 4 | rspcev 3554 | . . . 4 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
6 | 1, 2, 5 | syl2anc 584 | . . 3 ⊢ (𝐴 = ∅ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
7 | 6 | adantl 482 | . 2 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) ∧ 𝐴 = ∅) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
8 | neqne 2990 | . . . 4 ⊢ (¬ 𝐴 = ∅ → 𝐴 ≠ ∅) | |
9 | 8 | adantl 482 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) ∧ ¬ 𝐴 = ∅) → 𝐴 ≠ ∅) |
10 | simpll 763 | . . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → 𝐴 ⊆ ℝ) | |
11 | simplr 765 | . . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Fin) | |
12 | simpr 485 | . . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅) | |
13 | fiminre 11425 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) | |
14 | 10, 11, 12, 13 | syl3anc 1362 | . . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
15 | ssrexv 3950 | . . . 4 ⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)) | |
16 | 10, 14, 15 | sylc 65 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
17 | 9, 16 | syldan 591 | . 2 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
18 | 7, 17 | pm2.61dan 809 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ≠ wne 2982 ∀wral 3103 ∃wrex 3104 ⊆ wss 3854 ∅c0 4206 class class class wbr 4956 Fincfn 8347 ℝcr 10371 0cc0 10372 ≤ cle 10511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-resscn 10429 ax-1cn 10430 ax-addrcl 10433 ax-rnegex 10443 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-om 7428 df-1o 7944 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 |
This theorem is referenced by: infrefilb 41146 infxrrefi 41147 |
Copyright terms: Public domain | W3C validator |