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Mirrors > Home > ILE Home > Th. List > nninfisol | Unicode version |
Description: Finite elements of ℕ∞ are isolated. That is, given a natural number and any element of ℕ∞, it is decidable whether the natural number (when converted to an element of ℕ∞) is equal to the given element of ℕ∞. Stated in an online post by Martin Escardo. One way to understand this theorem is that you do not need to look at an unbounded number of elements of the sequence to decide whether it is equal to (in fact, you only need to look at two elements and tells you where to look). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.) |
Ref | Expression |
---|---|
nninfisol | ℕ∞ DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpllr 529 | . . . 4 ℕ∞ ℕ∞ | |
2 | simplr 525 | . . . 4 ℕ∞ | |
3 | simplll 528 | . . . 4 ℕ∞ | |
4 | simpr 109 | . . . 4 ℕ∞ | |
5 | 1, 2, 3, 4 | nninfisollem0 7106 | . . 3 ℕ∞ DECID |
6 | simp-4r 537 | . . . . 5 ℕ∞ ℕ∞ | |
7 | simpllr 529 | . . . . 5 ℕ∞ | |
8 | simp-4l 536 | . . . . 5 ℕ∞ | |
9 | simpr 109 | . . . . . . 7 ℕ∞ | |
10 | 9 | neqned 2347 | . . . . . 6 ℕ∞ |
11 | 10 | adantr 274 | . . . . 5 ℕ∞ |
12 | simpr 109 | . . . . 5 ℕ∞ | |
13 | 6, 7, 8, 11, 12 | nninfisollemne 7107 | . . . 4 ℕ∞ DECID |
14 | simp-4r 537 | . . . . 5 ℕ∞ ℕ∞ | |
15 | simpllr 529 | . . . . 5 ℕ∞ | |
16 | simp-4l 536 | . . . . 5 ℕ∞ | |
17 | 10 | adantr 274 | . . . . 5 ℕ∞ |
18 | simpr 109 | . . . . 5 ℕ∞ | |
19 | 14, 15, 16, 17, 18 | nninfisollemeq 7108 | . . . 4 ℕ∞ DECID |
20 | nninff 7099 | . . . . . . . . 9 ℕ∞ | |
21 | 20 | adantl 275 | . . . . . . . 8 ℕ∞ |
22 | nnpredcl 4607 | . . . . . . . . 9 | |
23 | 22 | adantr 274 | . . . . . . . 8 ℕ∞ |
24 | 21, 23 | ffvelrnd 5632 | . . . . . . 7 ℕ∞ |
25 | df2o3 6409 | . . . . . . 7 | |
26 | 24, 25 | eleqtrdi 2263 | . . . . . 6 ℕ∞ |
27 | elpri 3606 | . . . . . 6 | |
28 | 26, 27 | syl 14 | . . . . 5 ℕ∞ |
29 | 28 | ad2antrr 485 | . . . 4 ℕ∞ |
30 | 13, 19, 29 | mpjaodan 793 | . . 3 ℕ∞ DECID |
31 | nndceq0 4602 | . . . . 5 DECID | |
32 | exmiddc 831 | . . . . 5 DECID | |
33 | 31, 32 | syl 14 | . . . 4 |
34 | 33 | ad2antrr 485 | . . 3 ℕ∞ |
35 | 5, 30, 34 | mpjaodan 793 | . 2 ℕ∞ DECID |
36 | 1n0 6411 | . . . . . 6 | |
37 | 36 | neii 2342 | . . . . 5 |
38 | simpr 109 | . . . . . . . 8 ℕ∞ | |
39 | 38 | fveq1d 5498 | . . . . . . 7 ℕ∞ |
40 | eqid 2170 | . . . . . . . . . 10 | |
41 | eleq1 2233 | . . . . . . . . . . 11 | |
42 | 41 | ifbid 3547 | . . . . . . . . . 10 |
43 | id 19 | . . . . . . . . . 10 | |
44 | nnord 4596 | . . . . . . . . . . . . 13 | |
45 | ordirr 4526 | . . . . . . . . . . . . 13 | |
46 | 44, 45 | syl 14 | . . . . . . . . . . . 12 |
47 | 46 | iffalsed 3536 | . . . . . . . . . . 11 |
48 | peano1 4578 | . . . . . . . . . . 11 | |
49 | 47, 48 | eqeltrdi 2261 | . . . . . . . . . 10 |
50 | 40, 42, 43, 49 | fvmptd3 5589 | . . . . . . . . 9 |
51 | 50, 47 | eqtrd 2203 | . . . . . . . 8 |
52 | 51 | ad3antrrr 489 | . . . . . . 7 ℕ∞ |
53 | simplr 525 | . . . . . . 7 ℕ∞ | |
54 | 39, 52, 53 | 3eqtr3rd 2212 | . . . . . 6 ℕ∞ |
55 | 54 | ex 114 | . . . . 5 ℕ∞ |
56 | 37, 55 | mtoi 659 | . . . 4 ℕ∞ |
57 | 56 | olcd 729 | . . 3 ℕ∞ |
58 | df-dc 830 | . . 3 DECID | |
59 | 57, 58 | sylibr 133 | . 2 ℕ∞ DECID |
60 | simpl 108 | . . . . 5 ℕ∞ | |
61 | 21, 60 | ffvelrnd 5632 | . . . 4 ℕ∞ |
62 | 61, 25 | eleqtrdi 2263 | . . 3 ℕ∞ |
63 | elpri 3606 | . . 3 | |
64 | 62, 63 | syl 14 | . 2 ℕ∞ |
65 | 35, 59, 64 | mpjaodan 793 | 1 ℕ∞ DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 703 DECID wdc 829 wceq 1348 wcel 2141 wne 2340 c0 3414 cif 3526 cpr 3584 cuni 3796 cmpt 4050 word 4347 com 4574 wf 5194 cfv 5198 c1o 6388 c2o 6389 ℕ∞xnninf 7096 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1o 6395 df-2o 6396 df-map 6628 df-nninf 7097 |
This theorem is referenced by: (None) |
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