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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 524 | . . . 4 ℕ∞ ℕ∞ | |
2 | simplr 520 | . . . 4 ℕ∞ | |
3 | simplll 523 | . . . 4 ℕ∞ | |
4 | simpr 109 | . . . 4 ℕ∞ | |
5 | 1, 2, 3, 4 | nninfisollem0 7085 | . . 3 ℕ∞ DECID |
6 | simp-4r 532 | . . . . 5 ℕ∞ ℕ∞ | |
7 | simpllr 524 | . . . . 5 ℕ∞ | |
8 | simp-4l 531 | . . . . 5 ℕ∞ | |
9 | simpr 109 | . . . . . . 7 ℕ∞ | |
10 | 9 | neqned 2341 | . . . . . 6 ℕ∞ |
11 | 10 | adantr 274 | . . . . 5 ℕ∞ |
12 | simpr 109 | . . . . 5 ℕ∞ | |
13 | 6, 7, 8, 11, 12 | nninfisollemne 7086 | . . . 4 ℕ∞ DECID |
14 | simp-4r 532 | . . . . 5 ℕ∞ ℕ∞ | |
15 | simpllr 524 | . . . . 5 ℕ∞ | |
16 | simp-4l 531 | . . . . 5 ℕ∞ | |
17 | 10 | adantr 274 | . . . . 5 ℕ∞ |
18 | simpr 109 | . . . . 5 ℕ∞ | |
19 | 14, 15, 16, 17, 18 | nninfisollemeq 7087 | . . . 4 ℕ∞ DECID |
20 | nninff 7078 | . . . . . . . . 9 ℕ∞ | |
21 | 20 | adantl 275 | . . . . . . . 8 ℕ∞ |
22 | nnpredcl 4594 | . . . . . . . . 9 | |
23 | 22 | adantr 274 | . . . . . . . 8 ℕ∞ |
24 | 21, 23 | ffvelrnd 5615 | . . . . . . 7 ℕ∞ |
25 | df2o3 6389 | . . . . . . 7 | |
26 | 24, 25 | eleqtrdi 2257 | . . . . . 6 ℕ∞ |
27 | elpri 3593 | . . . . . 6 | |
28 | 26, 27 | syl 14 | . . . . 5 ℕ∞ |
29 | 28 | ad2antrr 480 | . . . 4 ℕ∞ |
30 | 13, 19, 29 | mpjaodan 788 | . . 3 ℕ∞ DECID |
31 | nndceq0 4589 | . . . . 5 DECID | |
32 | exmiddc 826 | . . . . 5 DECID | |
33 | 31, 32 | syl 14 | . . . 4 |
34 | 33 | ad2antrr 480 | . . 3 ℕ∞ |
35 | 5, 30, 34 | mpjaodan 788 | . 2 ℕ∞ DECID |
36 | 1n0 6391 | . . . . . 6 | |
37 | 36 | neii 2336 | . . . . 5 |
38 | simpr 109 | . . . . . . . 8 ℕ∞ | |
39 | 38 | fveq1d 5482 | . . . . . . 7 ℕ∞ |
40 | eqid 2164 | . . . . . . . . . 10 | |
41 | eleq1 2227 | . . . . . . . . . . 11 | |
42 | 41 | ifbid 3536 | . . . . . . . . . 10 |
43 | id 19 | . . . . . . . . . 10 | |
44 | nnord 4583 | . . . . . . . . . . . . 13 | |
45 | ordirr 4513 | . . . . . . . . . . . . 13 | |
46 | 44, 45 | syl 14 | . . . . . . . . . . . 12 |
47 | 46 | iffalsed 3525 | . . . . . . . . . . 11 |
48 | peano1 4565 | . . . . . . . . . . 11 | |
49 | 47, 48 | eqeltrdi 2255 | . . . . . . . . . 10 |
50 | 40, 42, 43, 49 | fvmptd3 5573 | . . . . . . . . 9 |
51 | 50, 47 | eqtrd 2197 | . . . . . . . 8 |
52 | 51 | ad3antrrr 484 | . . . . . . 7 ℕ∞ |
53 | simplr 520 | . . . . . . 7 ℕ∞ | |
54 | 39, 52, 53 | 3eqtr3rd 2206 | . . . . . 6 ℕ∞ |
55 | 54 | ex 114 | . . . . 5 ℕ∞ |
56 | 37, 55 | mtoi 654 | . . . 4 ℕ∞ |
57 | 56 | olcd 724 | . . 3 ℕ∞ |
58 | df-dc 825 | . . 3 DECID | |
59 | 57, 58 | sylibr 133 | . 2 ℕ∞ DECID |
60 | simpl 108 | . . . . 5 ℕ∞ | |
61 | 21, 60 | ffvelrnd 5615 | . . . 4 ℕ∞ |
62 | 61, 25 | eleqtrdi 2257 | . . 3 ℕ∞ |
63 | elpri 3593 | . . 3 | |
64 | 62, 63 | syl 14 | . 2 ℕ∞ |
65 | 35, 59, 64 | mpjaodan 788 | 1 ℕ∞ DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 DECID wdc 824 wceq 1342 wcel 2135 wne 2334 c0 3404 cif 3515 cpr 3571 cuni 3783 cmpt 4037 word 4334 com 4561 wf 5178 cfv 5182 c1o 6368 c2o 6369 ℕ∞xnninf 7075 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1o 6375 df-2o 6376 df-map 6607 df-nninf 7076 |
This theorem is referenced by: (None) |
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