| Mathbox for Jim Kingdon |
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| Mirrors > Home > ILE Home > Th. List > Mathboxes > nnnninfen | GIF version | ||
| Description: Equinumerosity of the natural numbers and ℕ∞ is equivalent to the Limited Principle of Omniscience (LPO). Remark in Section 1.1 of [Pradic2025], p. 2. (Contributed by Jim Kingdon, 8-Jul-2025.) |
| Ref | Expression |
|---|---|
| nnnninfen | ⊢ (ω ≈ ℕ∞ ↔ ω ∈ Omni) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nninfomni 16499 | . . 3 ⊢ ℕ∞ ∈ Omni | |
| 2 | enomni 7322 | . . 3 ⊢ (ω ≈ ℕ∞ → (ω ∈ Omni ↔ ℕ∞ ∈ Omni)) | |
| 3 | 1, 2 | mpbiri 168 | . 2 ⊢ (ω ≈ ℕ∞ → ω ∈ Omni) |
| 4 | lpowlpo 7351 | . . . . . 6 ⊢ (ω ∈ Omni → ω ∈ WOmni) | |
| 5 | nninfwlpo 7364 | . . . . . 6 ⊢ (∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 ↔ ω ∈ WOmni) | |
| 6 | 4, 5 | sylibr 134 | . . . . 5 ⊢ (ω ∈ Omni → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦) |
| 7 | nninfct 12583 | . . . . . 6 ⊢ (ω ∈ Omni → ∃𝑧 𝑧:ω–onto→(ℕ∞ ⊔ 1o)) | |
| 8 | infnninf 7307 | . . . . . . 7 ⊢ (𝑖 ∈ ω ↦ 1o) ∈ ℕ∞ | |
| 9 | elex2 2816 | . . . . . . 7 ⊢ ((𝑖 ∈ ω ↦ 1o) ∈ ℕ∞ → ∃𝑗 𝑗 ∈ ℕ∞) | |
| 10 | ctm 7292 | . . . . . . 7 ⊢ (∃𝑗 𝑗 ∈ ℕ∞ → (∃𝑧 𝑧:ω–onto→(ℕ∞ ⊔ 1o) ↔ ∃𝑧 𝑧:ω–onto→ℕ∞)) | |
| 11 | 8, 9, 10 | mp2b 8 | . . . . . 6 ⊢ (∃𝑧 𝑧:ω–onto→(ℕ∞ ⊔ 1o) ↔ ∃𝑧 𝑧:ω–onto→ℕ∞) |
| 12 | 7, 11 | sylib 122 | . . . . 5 ⊢ (ω ∈ Omni → ∃𝑧 𝑧:ω–onto→ℕ∞) |
| 13 | nninfinf 10682 | . . . . . 6 ⊢ ω ≼ ℕ∞ | |
| 14 | 13 | a1i 9 | . . . . 5 ⊢ (ω ∈ Omni → ω ≼ ℕ∞) |
| 15 | ctinf 13022 | . . . . 5 ⊢ (ℕ∞ ≈ ℕ ↔ (∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 ∧ ∃𝑧 𝑧:ω–onto→ℕ∞ ∧ ω ≼ ℕ∞)) | |
| 16 | 6, 12, 14, 15 | syl3anbrc 1205 | . . . 4 ⊢ (ω ∈ Omni → ℕ∞ ≈ ℕ) |
| 17 | nnenom 10673 | . . . 4 ⊢ ℕ ≈ ω | |
| 18 | entr 6949 | . . . 4 ⊢ ((ℕ∞ ≈ ℕ ∧ ℕ ≈ ω) → ℕ∞ ≈ ω) | |
| 19 | 16, 17, 18 | sylancl 413 | . . 3 ⊢ (ω ∈ Omni → ℕ∞ ≈ ω) |
| 20 | 19 | ensymd 6948 | . 2 ⊢ (ω ∈ Omni → ω ≈ ℕ∞) |
| 21 | 3, 20 | impbii 126 | 1 ⊢ (ω ≈ ℕ∞ ↔ ω ∈ Omni) |
| Colors of variables: wff set class |
| Syntax hints: ↔ wb 105 DECID wdc 839 ∃wex 1538 ∈ wcel 2200 ∀wral 2508 class class class wbr 4083 ↦ cmpt 4145 ωcom 4683 –onto→wfo 5319 1oc1o 6566 ≈ cen 6898 ≼ cdom 6899 ⊔ cdju 7220 ℕ∞xnninf 7302 Omnicomni 7317 WOmnicwomni 7346 ℕcn 9126 |
| 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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4259 ax-pr 4294 ax-un 4525 ax-setind 4630 ax-iinf 4681 ax-cnex 8106 ax-resscn 8107 ax-1cn 8108 ax-1re 8109 ax-icn 8110 ax-addcl 8111 ax-addrcl 8112 ax-mulcl 8113 ax-addcom 8115 ax-addass 8117 ax-distr 8119 ax-i2m1 8120 ax-0lt1 8121 ax-0id 8123 ax-rnegex 8124 ax-cnre 8126 ax-pre-ltirr 8127 ax-pre-ltwlin 8128 ax-pre-lttrn 8129 ax-pre-apti 8130 ax-pre-ltadd 8131 |
| This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4385 df-po 4388 df-iso 4389 df-iord 4458 df-on 4460 df-ilim 4461 df-suc 4463 df-iom 4684 df-xp 4726 df-rel 4727 df-cnv 4728 df-co 4729 df-dm 4730 df-rn 4731 df-res 4732 df-ima 4733 df-iota 5281 df-fun 5323 df-fn 5324 df-f 5325 df-f1 5326 df-fo 5327 df-f1o 5328 df-fv 5329 df-isom 5330 df-riota 5963 df-ov 6013 df-oprab 6014 df-mpo 6015 df-1st 6295 df-2nd 6296 df-recs 6462 df-frec 6548 df-1o 6573 df-2o 6574 df-er 6693 df-map 6810 df-pm 6811 df-en 6901 df-dom 6902 df-fin 6903 df-sup 7167 df-inf 7168 df-dju 7221 df-inl 7230 df-inr 7231 df-case 7267 df-nninf 7303 df-omni 7318 df-markov 7335 df-womni 7347 df-pnf 8199 df-mnf 8200 df-xr 8201 df-ltxr 8202 df-le 8203 df-sub 8335 df-neg 8336 df-inn 9127 df-n0 9386 df-xnn0 9449 df-z 9463 df-uz 9739 df-fz 10222 df-fzo 10356 df-seqfrec 10687 |
| This theorem is referenced by: (None) |
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