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Mirrors > Home > ILE Home > Th. List > Mathboxes > nninfsel | GIF version |
Description: 𝐸 is a selection function for ℕ∞. Theorem 3.6 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 9-Aug-2022.) |
Ref | Expression |
---|---|
nninfsel.e | ⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))) |
nninfsel.q | ⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞)) |
nninfsel.1 | ⊢ (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o) |
Ref | Expression |
---|---|
nninfsel | ⊢ (𝜑 → ∀𝑝 ∈ ℕ∞ (𝑄‘𝑝) = 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nninfsel.q | . 2 ⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞)) | |
2 | nninfsel.e | . . . . 5 ⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))) | |
3 | nninfsel.1 | . . . . 5 ⊢ (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o) | |
4 | 2, 1, 3 | nninfsellemeqinf 13201 | . . . 4 ⊢ (𝜑 → (𝐸‘𝑄) = (𝑖 ∈ ω ↦ 1o)) |
5 | 4 | fveq2d 5418 | . . 3 ⊢ (𝜑 → (𝑄‘(𝐸‘𝑄)) = (𝑄‘(𝑖 ∈ ω ↦ 1o))) |
6 | 5, 3 | eqtr3d 2172 | . 2 ⊢ (𝜑 → (𝑄‘(𝑖 ∈ ω ↦ 1o)) = 1o) |
7 | 1 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑄 ∈ (2o ↑𝑚 ℕ∞)) |
8 | 3 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → (𝑄‘(𝐸‘𝑄)) = 1o) |
9 | simpr 109 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 𝑥 ∈ ω) | |
10 | 2, 7, 8, 9 | nninfsellemqall 13200 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑥, 1o, ∅))) = 1o) |
11 | 10 | ralrimiva 2503 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑥, 1o, ∅))) = 1o) |
12 | 1, 6, 11 | nninfall 13193 | 1 ⊢ (𝜑 → ∀𝑝 ∈ ℕ∞ (𝑄‘𝑝) = 1o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∀wral 2414 ∅c0 3358 ifcif 3469 ↦ cmpt 3984 suc csuc 4282 ωcom 4499 ‘cfv 5118 (class class class)co 5767 1oc1o 6299 2oc2o 6300 ↑𝑚 cmap 6535 ℕ∞xnninf 6998 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1o 6306 df-2o 6307 df-map 6537 df-nninf 7000 |
This theorem is referenced by: nninfomnilem 13203 |
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