![]() |
Mathbox for Jim Kingdon |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > Mathboxes > nninfomnilem | GIF version |
Description: Lemma for nninfomni 12214. (Contributed by Jim Kingdon, 10-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, ∅))) |
Ref | Expression |
---|---|
nninfomnilem | ⊢ ℕ∞ ∈ Omni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nninfex 12204 | . . 3 ⊢ ℕ∞ ∈ V | |
2 | isomnimap 6856 | . . 3 ⊢ (ℕ∞ ∈ V → (ℕ∞ ∈ Omni ↔ ∀𝑟 ∈ (2o ↑𝑚 ℕ∞)(∃𝑝 ∈ ℕ∞ (𝑟‘𝑝) = ∅ ∨ ∀𝑝 ∈ ℕ∞ (𝑟‘𝑝) = 1o))) | |
3 | 1, 2 | ax-mp 7 | . 2 ⊢ (ℕ∞ ∈ Omni ↔ ∀𝑟 ∈ (2o ↑𝑚 ℕ∞)(∃𝑝 ∈ ℕ∞ (𝑟‘𝑝) = ∅ ∨ ∀𝑝 ∈ ℕ∞ (𝑟‘𝑝) = 1o)) |
4 | elmapi 6443 | . . . . . 6 ⊢ (𝑟 ∈ (2o ↑𝑚 ℕ∞) → 𝑟:ℕ∞⟶2o) | |
5 | nninfsel.e | . . . . . . . 8 ⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))) | |
6 | 5 | nninfself 12208 | . . . . . . 7 ⊢ 𝐸:(2o ↑𝑚 ℕ∞)⟶ℕ∞ |
7 | 6 | ffvelrni 5449 | . . . . . 6 ⊢ (𝑟 ∈ (2o ↑𝑚 ℕ∞) → (𝐸‘𝑟) ∈ ℕ∞) |
8 | 4, 7 | ffvelrnd 5451 | . . . . 5 ⊢ (𝑟 ∈ (2o ↑𝑚 ℕ∞) → (𝑟‘(𝐸‘𝑟)) ∈ 2o) |
9 | df2o3 6211 | . . . . 5 ⊢ 2o = {∅, 1o} | |
10 | 8, 9 | syl6eleq 2181 | . . . 4 ⊢ (𝑟 ∈ (2o ↑𝑚 ℕ∞) → (𝑟‘(𝐸‘𝑟)) ∈ {∅, 1o}) |
11 | elpri 3475 | . . . 4 ⊢ ((𝑟‘(𝐸‘𝑟)) ∈ {∅, 1o} → ((𝑟‘(𝐸‘𝑟)) = ∅ ∨ (𝑟‘(𝐸‘𝑟)) = 1o)) | |
12 | 10, 11 | syl 14 | . . 3 ⊢ (𝑟 ∈ (2o ↑𝑚 ℕ∞) → ((𝑟‘(𝐸‘𝑟)) = ∅ ∨ (𝑟‘(𝐸‘𝑟)) = 1o)) |
13 | fveq2 5320 | . . . . . . . 8 ⊢ (𝑝 = (𝐸‘𝑟) → (𝑟‘𝑝) = (𝑟‘(𝐸‘𝑟))) | |
14 | 13 | eqeq1d 2097 | . . . . . . 7 ⊢ (𝑝 = (𝐸‘𝑟) → ((𝑟‘𝑝) = ∅ ↔ (𝑟‘(𝐸‘𝑟)) = ∅)) |
15 | 14 | rspcev 2725 | . . . . . 6 ⊢ (((𝐸‘𝑟) ∈ ℕ∞ ∧ (𝑟‘(𝐸‘𝑟)) = ∅) → ∃𝑝 ∈ ℕ∞ (𝑟‘𝑝) = ∅) |
16 | 15 | ex 114 | . . . . 5 ⊢ ((𝐸‘𝑟) ∈ ℕ∞ → ((𝑟‘(𝐸‘𝑟)) = ∅ → ∃𝑝 ∈ ℕ∞ (𝑟‘𝑝) = ∅)) |
17 | 7, 16 | syl 14 | . . . 4 ⊢ (𝑟 ∈ (2o ↑𝑚 ℕ∞) → ((𝑟‘(𝐸‘𝑟)) = ∅ → ∃𝑝 ∈ ℕ∞ (𝑟‘𝑝) = ∅)) |
18 | simpl 108 | . . . . . 6 ⊢ ((𝑟 ∈ (2o ↑𝑚 ℕ∞) ∧ (𝑟‘(𝐸‘𝑟)) = 1o) → 𝑟 ∈ (2o ↑𝑚 ℕ∞)) | |
19 | simpr 109 | . . . . . 6 ⊢ ((𝑟 ∈ (2o ↑𝑚 ℕ∞) ∧ (𝑟‘(𝐸‘𝑟)) = 1o) → (𝑟‘(𝐸‘𝑟)) = 1o) | |
20 | 5, 18, 19 | nninfsel 12212 | . . . . 5 ⊢ ((𝑟 ∈ (2o ↑𝑚 ℕ∞) ∧ (𝑟‘(𝐸‘𝑟)) = 1o) → ∀𝑝 ∈ ℕ∞ (𝑟‘𝑝) = 1o) |
21 | 20 | ex 114 | . . . 4 ⊢ (𝑟 ∈ (2o ↑𝑚 ℕ∞) → ((𝑟‘(𝐸‘𝑟)) = 1o → ∀𝑝 ∈ ℕ∞ (𝑟‘𝑝) = 1o)) |
22 | 17, 21 | orim12d 736 | . . 3 ⊢ (𝑟 ∈ (2o ↑𝑚 ℕ∞) → (((𝑟‘(𝐸‘𝑟)) = ∅ ∨ (𝑟‘(𝐸‘𝑟)) = 1o) → (∃𝑝 ∈ ℕ∞ (𝑟‘𝑝) = ∅ ∨ ∀𝑝 ∈ ℕ∞ (𝑟‘𝑝) = 1o))) |
23 | 12, 22 | mpd 13 | . 2 ⊢ (𝑟 ∈ (2o ↑𝑚 ℕ∞) → (∃𝑝 ∈ ℕ∞ (𝑟‘𝑝) = ∅ ∨ ∀𝑝 ∈ ℕ∞ (𝑟‘𝑝) = 1o)) |
24 | 3, 23 | mprgbir 2434 | 1 ⊢ ℕ∞ ∈ Omni |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 665 = wceq 1290 ∈ wcel 1439 ∀wral 2360 ∃wrex 2361 Vcvv 2622 ∅c0 3289 ifcif 3399 {cpr 3453 ↦ cmpt 3907 suc csuc 4203 ωcom 4420 ‘cfv 5030 (class class class)co 5668 1oc1o 6190 2oc2o 6191 ↑𝑚 cmap 6421 Omnicomni 6851 ℕ∞xnninf 6852 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3962 ax-sep 3965 ax-nul 3973 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-iinf 4418 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2624 df-sbc 2844 df-csb 2937 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-if 3400 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-iun 3740 df-br 3854 df-opab 3908 df-mpt 3909 df-tr 3945 df-id 4131 df-iord 4204 df-on 4206 df-suc 4209 df-iom 4421 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-f1 5035 df-fo 5036 df-f1o 5037 df-fv 5038 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-1o 6197 df-2o 6198 df-map 6423 df-omni 6853 df-nninf 6854 |
This theorem is referenced by: nninfomni 12214 |
Copyright terms: Public domain | W3C validator |