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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 7074 | . . 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 2334 | . . . . . 6 ℕ∞ |
11 | 10 | adantr 274 | . . . . 5 ℕ∞ |
12 | simpr 109 | . . . . 5 ℕ∞ | |
13 | 6, 7, 8, 11, 12 | nninfisollemne 7075 | . . . 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 7076 | . . . 4 ℕ∞ DECID |
20 | nninff 7067 | . . . . . . . . 9 ℕ∞ | |
21 | 20 | adantl 275 | . . . . . . . 8 ℕ∞ |
22 | nnpredcl 4583 | . . . . . . . . 9 | |
23 | 22 | adantr 274 | . . . . . . . 8 ℕ∞ |
24 | 21, 23 | ffvelrnd 5604 | . . . . . . 7 ℕ∞ |
25 | df2o3 6378 | . . . . . . 7 | |
26 | 24, 25 | eleqtrdi 2250 | . . . . . 6 ℕ∞ |
27 | elpri 3583 | . . . . . 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 4578 | . . . . 5 DECID | |
32 | exmiddc 822 | . . . . 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 6380 | . . . . . 6 | |
37 | 36 | neii 2329 | . . . . 5 |
38 | simpr 109 | . . . . . . . 8 ℕ∞ | |
39 | 38 | fveq1d 5471 | . . . . . . 7 ℕ∞ |
40 | eqid 2157 | . . . . . . . . . 10 | |
41 | eleq1 2220 | . . . . . . . . . . 11 | |
42 | 41 | ifbid 3526 | . . . . . . . . . 10 |
43 | id 19 | . . . . . . . . . 10 | |
44 | nnord 4572 | . . . . . . . . . . . . 13 | |
45 | ordirr 4502 | . . . . . . . . . . . . 13 | |
46 | 44, 45 | syl 14 | . . . . . . . . . . . 12 |
47 | 46 | iffalsed 3515 | . . . . . . . . . . 11 |
48 | peano1 4554 | . . . . . . . . . . 11 | |
49 | 47, 48 | eqeltrdi 2248 | . . . . . . . . . 10 |
50 | 40, 42, 43, 49 | fvmptd3 5562 | . . . . . . . . 9 |
51 | 50, 47 | eqtrd 2190 | . . . . . . . 8 |
52 | 51 | ad3antrrr 484 | . . . . . . 7 ℕ∞ |
53 | simplr 520 | . . . . . . 7 ℕ∞ | |
54 | 39, 52, 53 | 3eqtr3rd 2199 | . . . . . 6 ℕ∞ |
55 | 54 | ex 114 | . . . . 5 ℕ∞ |
56 | 37, 55 | mtoi 654 | . . . 4 ℕ∞ |
57 | 56 | olcd 724 | . . 3 ℕ∞ |
58 | df-dc 821 | . . 3 DECID | |
59 | 57, 58 | sylibr 133 | . 2 ℕ∞ DECID |
60 | simpl 108 | . . . . 5 ℕ∞ | |
61 | 21, 60 | ffvelrnd 5604 | . . . 4 ℕ∞ |
62 | 61, 25 | eleqtrdi 2250 | . . 3 ℕ∞ |
63 | elpri 3583 | . . 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 820 wceq 1335 wcel 2128 wne 2327 c0 3394 cif 3505 cpr 3561 cuni 3773 cmpt 4026 word 4323 com 4550 wf 5167 cfv 5171 c1o 6357 c2o 6358 ℕ∞xnninf 7064 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1o 6364 df-2o 6365 df-map 6596 df-nninf 7065 |
This theorem is referenced by: (None) |
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