Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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.) |
Ref | Expression |
---|---|
fimaxre | ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltso 9997 | . . . 4 ⊢ < Or ℝ | |
2 | soss 4977 | . . . 4 ⊢ (𝐴 ⊆ ℝ → ( < Or ℝ → < Or 𝐴)) | |
3 | 1, 2 | mpi 20 | . . 3 ⊢ (𝐴 ⊆ ℝ → < Or 𝐴) |
4 | fimaxg 8092 | . . 3 ⊢ (( < Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦 < 𝑥)) | |
5 | 3, 4 | syl3an1 1351 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦 < 𝑥)) |
6 | ssel 3562 | . . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ → (𝑥 ∈ 𝐴 → 𝑥 ∈ ℝ)) | |
7 | ssel 3562 | . . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ)) | |
8 | 6, 7 | anim12d 584 | . . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ))) |
9 | 8 | imp 444 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) |
10 | leloe 10003 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑦 = 𝑥))) | |
11 | 10 | ancoms 468 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑦 = 𝑥))) |
12 | equcom 1932 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
13 | 12 | orbi2i 540 | . . . . . . . . . 10 ⊢ ((𝑦 < 𝑥 ∨ 𝑦 = 𝑥) ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)) |
14 | orcom 401 | . . . . . . . . . 10 ⊢ ((𝑦 < 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 = 𝑦 ∨ 𝑦 < 𝑥)) | |
15 | neor 2873 | . . . . . . . . . 10 ⊢ ((𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ (𝑥 ≠ 𝑦 → 𝑦 < 𝑥)) | |
16 | 13, 14, 15 | 3bitri 285 | . . . . . . . . 9 ⊢ ((𝑦 < 𝑥 ∨ 𝑦 = 𝑥) ↔ (𝑥 ≠ 𝑦 → 𝑦 < 𝑥)) |
17 | 11, 16 | syl6bb 275 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 ≤ 𝑥 ↔ (𝑥 ≠ 𝑦 → 𝑦 < 𝑥))) |
18 | 17 | biimprd 237 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 ≠ 𝑦 → 𝑦 < 𝑥) → 𝑦 ≤ 𝑥)) |
19 | 9, 18 | syl 17 | . . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 ≠ 𝑦 → 𝑦 < 𝑥) → 𝑦 ≤ 𝑥)) |
20 | 19 | anassrs 678 | . . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑥 ≠ 𝑦 → 𝑦 < 𝑥) → 𝑦 ≤ 𝑥)) |
21 | 20 | ralimdva 2945 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦 < 𝑥) → ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
22 | 21 | reximdva 3000 | . . 3 ⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦 < 𝑥) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
23 | 22 | 3ad2ant1 1075 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦 < 𝑥) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
24 | 5, 23 | mpd 15 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 Or wor 4958 Fincfn 7841 ℝcr 9814 < clt 9953 ≤ cle 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 |
This theorem is referenced by: fimaxre2 10848 fiminre 10851 0ram2 15563 0ramcl 15565 prmgaplem3 15595 ballotlemfc0 29881 ballotlemfcc 29882 filbcmb 32705 |
Copyright terms: Public domain | W3C validator |