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Mirrors > Home > ILE Home > Th. List > nninfdcinf | GIF version |
Description: The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of ℕ∞ equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.) |
Ref | Expression |
---|---|
nninfdcinf.w | ⊢ (𝜑 → ω ∈ WOmni) |
nninfdcinf.n | ⊢ (𝜑 → 𝑁 ∈ ℕ∞) |
Ref | Expression |
---|---|
nninfdcinf | ⊢ (𝜑 → DECID 𝑁 = (𝑖 ∈ ω ↦ 1o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 5509 | . . . . . 6 ⊢ (𝑓 = 𝑁 → (𝑓‘𝑥) = (𝑁‘𝑥)) | |
2 | 1 | eqeq1d 2186 | . . . . 5 ⊢ (𝑓 = 𝑁 → ((𝑓‘𝑥) = 1o ↔ (𝑁‘𝑥) = 1o)) |
3 | 2 | ralbidv 2477 | . . . 4 ⊢ (𝑓 = 𝑁 → (∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o)) |
4 | 3 | dcbid 838 | . . 3 ⊢ (𝑓 = 𝑁 → (DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ DECID ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o)) |
5 | nninfdcinf.w | . . . 4 ⊢ (𝜑 → ω ∈ WOmni) | |
6 | 5 | elexd 2750 | . . . . 5 ⊢ (𝜑 → ω ∈ V) |
7 | iswomnimap 7157 | . . . . 5 ⊢ (ω ∈ V → (ω ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o)) | |
8 | 6, 7 | syl 14 | . . . 4 ⊢ (𝜑 → (ω ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o)) |
9 | 5, 8 | mpbid 147 | . . 3 ⊢ (𝜑 → ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o) |
10 | nninfdcinf.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ∞) | |
11 | nninff 7114 | . . . . 5 ⊢ (𝑁 ∈ ℕ∞ → 𝑁:ω⟶2o) | |
12 | 10, 11 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑁:ω⟶2o) |
13 | 2onn 6515 | . . . . . 6 ⊢ 2o ∈ ω | |
14 | 13 | elexi 2749 | . . . . 5 ⊢ 2o ∈ V |
15 | omex 4588 | . . . . 5 ⊢ ω ∈ V | |
16 | 14, 15 | elmap 6670 | . . . 4 ⊢ (𝑁 ∈ (2o ↑𝑚 ω) ↔ 𝑁:ω⟶2o) |
17 | 12, 16 | sylibr 134 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (2o ↑𝑚 ω)) |
18 | 4, 9, 17 | rspcdva 2846 | . 2 ⊢ (𝜑 → DECID ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o) |
19 | 12 | ffnd 5361 | . . . 4 ⊢ (𝜑 → 𝑁 Fn ω) |
20 | eqidd 2178 | . . . 4 ⊢ (𝑥 = 𝑖 → 1o = 1o) | |
21 | 1onn 6514 | . . . . 5 ⊢ 1o ∈ ω | |
22 | 21 | a1i 9 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 1o ∈ ω) |
23 | 21 | a1i 9 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → 1o ∈ ω) |
24 | 19, 20, 22, 23 | fnmptfvd 5615 | . . 3 ⊢ (𝜑 → (𝑁 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o)) |
25 | 24 | dcbid 838 | . 2 ⊢ (𝜑 → (DECID 𝑁 = (𝑖 ∈ ω ↦ 1o) ↔ DECID ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o)) |
26 | 18, 25 | mpbird 167 | 1 ⊢ (𝜑 → DECID 𝑁 = (𝑖 ∈ ω ↦ 1o)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 834 = wceq 1353 ∈ wcel 2148 ∀wral 2455 Vcvv 2737 ↦ cmpt 4061 ωcom 4585 ⟶wf 5207 ‘cfv 5211 (class class class)co 5868 1oc1o 6403 2oc2o 6404 ↑𝑚 cmap 6641 ℕ∞xnninf 7111 WOmnicwomni 7154 |
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-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 |
This theorem depends on definitions: df-bi 117 df-dc 835 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-ral 2460 df-rex 2461 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-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1o 6410 df-2o 6411 df-map 6643 df-nninf 7112 df-womni 7155 |
This theorem is referenced by: (None) |
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