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| Mirrors > Home > MPE Home > Th. List > fimaxre | Structured version Visualization version GIF version | ||
| Description: A finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Steven Nguyen, 3-Jun-2023.) |
| Ref | Expression |
|---|---|
| fimaxre | ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ltso 11361 | . . . 4 ⊢ < Or ℝ | |
| 2 | soss 5575 | . . . 4 ⊢ (𝐴 ⊆ ℝ → ( < Or ℝ → < Or 𝐴)) | |
| 3 | 1, 2 | mpi 21 | . . 3 ⊢ (𝐴 ⊆ ℝ → < Or 𝐴) |
| 4 | fimaxg 9256 | . . 3 ⊢ (( < Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦 < 𝑥)) | |
| 5 | 3, 4 | syl3an1 1181 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦 < 𝑥)) |
| 6 | ssel2 3925 | . . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ) | |
| 7 | 6 | adantrl 729 | . . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℝ) |
| 8 | ssel2 3925 | . . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) | |
| 9 | 8 | adantrr 730 | . . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℝ) |
| 10 | 7, 9 | leloed 11424 | . . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑦 = 𝑥))) |
| 11 | orcom 884 | . . . . . . . . . 10 ⊢ ((𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)) | |
| 12 | equcom 2051 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
| 13 | 12 | orbi2i 926 | . . . . . . . . . 10 ⊢ ((𝑦 < 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) |
| 14 | 11, 13 | bitri 278 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) |
| 15 | 14 | a1i 11 | . . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ (𝑦 < 𝑥 ∨ 𝑦 = 𝑥))) |
| 16 | neor 3047 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ (𝑥 ≠ 𝑦 → 𝑦 < 𝑥)) | |
| 17 | 16 | a1i 11 | . . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ (𝑥 ≠ 𝑦 → 𝑦 < 𝑥))) |
| 18 | 10, 15, 17 | 3bitr2d 310 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑦 ≤ 𝑥 ↔ (𝑥 ≠ 𝑦 → 𝑦 < 𝑥))) |
| 19 | 18 | biimprd 251 | . . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 ≠ 𝑦 → 𝑦 < 𝑥) → 𝑦 ≤ 𝑥)) |
| 20 | 19 | anassrs 473 | . . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑥 ≠ 𝑦 → 𝑦 < 𝑥) → 𝑦 ≤ 𝑥)) |
| 21 | 20 | ralimdva 3174 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦 < 𝑥) → ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
| 22 | 21 | reximdva 3175 | . . 3 ⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦 < 𝑥) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
| 23 | 22 | 3ad2ant1 1151 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦 < 𝑥) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
| 24 | 5, 23 | mpd 16 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ⊆ wss 3898 ∅c0 4278 class class class wbr 5102 Or wor 5554 Fincfn 8951 ℝcr 11170 < clt 11314 ≤ cle 11315 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-resscn 11228 ax-pre-lttri 11245 ax-pre-lttrn 11246 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-om 7861 df-er 8695 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-pnf 11316 df-mnf 11317 df-xr 11318 df-ltxr 11319 df-le 11320 |
| This theorem is used by: fimaxre2 12231 0ram2 17160 0ramcl 17162 prmgaplem3 17192 ballotlemfc0 35059 ballotlemfcc 35060 filbcmb 38594 |
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