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Mirrors > Home > ILE Home > Th. List > fimaxq | GIF version |
Description: A finite set of rational numbers has a maximum. (Contributed by Jim Kingdon, 6-Sep-2022.) |
Ref | Expression |
---|---|
fimaxq | ⊢ ((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qssre 9449 | . . . . 5 ⊢ ℚ ⊆ ℝ | |
2 | sstr 3110 | . . . . . 6 ⊢ ((𝐴 ⊆ ℚ ∧ ℚ ⊆ ℝ) → 𝐴 ⊆ ℝ) | |
3 | ltso 7866 | . . . . . . 7 ⊢ < Or ℝ | |
4 | sopo 4243 | . . . . . . 7 ⊢ ( < Or ℝ → < Po ℝ) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ < Po ℝ |
6 | poss 4228 | . . . . . 6 ⊢ (𝐴 ⊆ ℝ → ( < Po ℝ → < Po 𝐴)) | |
7 | 2, 5, 6 | mpisyl 1423 | . . . . 5 ⊢ ((𝐴 ⊆ ℚ ∧ ℚ ⊆ ℝ) → < Po 𝐴) |
8 | 1, 7 | mpan2 422 | . . . 4 ⊢ (𝐴 ⊆ ℚ → < Po 𝐴) |
9 | 8 | 3ad2ant1 1003 | . . 3 ⊢ ((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → < Po 𝐴) |
10 | simpl1 985 | . . . . . 6 ⊢ (((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐴 ⊆ ℚ) | |
11 | simprl 521 | . . . . . 6 ⊢ (((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴) | |
12 | 10, 11 | sseldd 3103 | . . . . 5 ⊢ (((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℚ) |
13 | simprr 522 | . . . . . 6 ⊢ (((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴) | |
14 | 10, 13 | sseldd 3103 | . . . . 5 ⊢ (((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℚ) |
15 | qtri3or 10051 | . . . . 5 ⊢ ((𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)) | |
16 | 12, 14, 15 | syl2anc 409 | . . . 4 ⊢ (((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)) |
17 | 16 | ralrimivva 2517 | . . 3 ⊢ ((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)) |
18 | simp2 983 | . . 3 ⊢ ((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Fin) | |
19 | simp3 984 | . . 3 ⊢ ((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅) | |
20 | 9, 17, 18, 19 | fimax2gtri 6803 | . 2 ⊢ ((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦) |
21 | simpll1 1021 | . . . . . . 7 ⊢ ((((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐴 ⊆ ℚ) | |
22 | simpr 109 | . . . . . . 7 ⊢ ((((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
23 | 21, 22 | sseldd 3103 | . . . . . 6 ⊢ ((((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℚ) |
24 | qre 9444 | . . . . . 6 ⊢ (𝑦 ∈ ℚ → 𝑦 ∈ ℝ) | |
25 | 23, 24 | syl 14 | . . . . 5 ⊢ ((((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ) |
26 | simplr 520 | . . . . . . 7 ⊢ ((((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
27 | 21, 26 | sseldd 3103 | . . . . . 6 ⊢ ((((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ ℚ) |
28 | qre 9444 | . . . . . 6 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ) | |
29 | 27, 28 | syl 14 | . . . . 5 ⊢ ((((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ ℝ) |
30 | 25, 29 | lenltd 7904 | . . . 4 ⊢ ((((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ 𝑥 ↔ ¬ 𝑥 < 𝑦)) |
31 | 30 | ralbidva 2434 | . . 3 ⊢ (((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦)) |
32 | 31 | rexbidva 2435 | . 2 ⊢ ((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦)) |
33 | 20, 32 | mpbird 166 | 1 ⊢ ((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ w3o 962 ∧ w3a 963 ∈ wcel 1481 ≠ wne 2309 ∀wral 2417 ∃wrex 2418 ⊆ wss 3076 ∅c0 3368 class class class wbr 3937 Po wpo 4224 Or wor 4225 Fincfn 6642 ℝcr 7643 < clt 7824 ≤ cle 7825 ℚcq 9438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-er 6437 df-en 6643 df-fin 6645 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-n0 9002 df-z 9079 df-q 9439 df-rp 9471 |
This theorem is referenced by: zfz1iso 10616 |
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