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Mirrors > Home > ILE Home > Th. List > nnwodc | GIF version |
Description: Well-ordering principle: any inhabited decidable set of positive integers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34. (Contributed by NM, 17-Aug-2001.) (Revised by Jim Kingdon, 23-Oct-2024.) |
Ref | Expression |
---|---|
nnwodc | ⊢ ((𝐴 ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnmindc 12018 | . . 3 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴 ∧ ∃𝑤 𝑤 ∈ 𝐴) → inf(𝐴, ℝ, < ) ∈ 𝐴) | |
2 | 1 | 3com23 1209 | . 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴) → inf(𝐴, ℝ, < ) ∈ 𝐴) |
3 | simpl1 1000 | . . . 4 ⊢ (((𝐴 ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐴 ⊆ ℕ) | |
4 | simpl3 1002 | . . . 4 ⊢ (((𝐴 ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴) | |
5 | simpr 110 | . . . 4 ⊢ (((𝐴 ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
6 | nnminle 12019 | . . . 4 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝑦) | |
7 | 3, 4, 5, 6 | syl3anc 1238 | . . 3 ⊢ (((𝐴 ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝑦) |
8 | 7 | ralrimiva 2550 | . 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 inf(𝐴, ℝ, < ) ≤ 𝑦) |
9 | breq1 4003 | . . . 4 ⊢ (𝑥 = inf(𝐴, ℝ, < ) → (𝑥 ≤ 𝑦 ↔ inf(𝐴, ℝ, < ) ≤ 𝑦)) | |
10 | 9 | ralbidv 2477 | . . 3 ⊢ (𝑥 = inf(𝐴, ℝ, < ) → (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝐴 inf(𝐴, ℝ, < ) ≤ 𝑦)) |
11 | 10 | rspcev 2841 | . 2 ⊢ ((inf(𝐴, ℝ, < ) ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 inf(𝐴, ℝ, < ) ≤ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
12 | 2, 8, 11 | syl2anc 411 | 1 ⊢ ((𝐴 ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 DECID wdc 834 ∧ w3a 978 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 ⊆ wss 3129 class class class wbr 4000 infcinf 6976 ℝcr 7801 < clt 7982 ≤ cle 7983 ℕcn 8908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-addcom 7902 ax-addass 7904 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-0id 7910 ax-rnegex 7911 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-po 4293 df-iso 4294 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-sup 6977 df-inf 6978 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-inn 8909 df-n0 9166 df-z 9243 df-uz 9518 df-fz 9996 df-fzo 10129 |
This theorem is referenced by: uzwodc 12021 nnwofdc 12022 |
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