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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 ![]() ![]() ![]() |
Ref | Expression |
---|---|
nninfisol |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpllr 534 |
. . . 4
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2 | simplr 528 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | simplll 533 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | simpr 110 |
. . . 4
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5 | 1, 2, 3, 4 | nninfisollem0 7141 |
. . 3
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6 | simp-4r 542 |
. . . . 5
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7 | simpllr 534 |
. . . . 5
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8 | simp-4l 541 |
. . . . 5
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9 | simpr 110 |
. . . . . . 7
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10 | 9 | neqned 2364 |
. . . . . 6
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11 | 10 | adantr 276 |
. . . . 5
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12 | simpr 110 |
. . . . 5
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13 | 6, 7, 8, 11, 12 | nninfisollemne 7142 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | simp-4r 542 |
. . . . 5
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15 | simpllr 534 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | simp-4l 541 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 10 | adantr 276 |
. . . . 5
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18 | simpr 110 |
. . . . 5
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19 | 14, 15, 16, 17, 18 | nninfisollemeq 7143 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | nninff 7134 |
. . . . . . . . 9
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21 | 20 | adantl 277 |
. . . . . . . 8
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22 | nnpredcl 4634 |
. . . . . . . . 9
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23 | 22 | adantr 276 |
. . . . . . . 8
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24 | 21, 23 | ffvelcdmd 5665 |
. . . . . . 7
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25 | df2o3 6444 |
. . . . . . 7
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26 | 24, 25 | eleqtrdi 2280 |
. . . . . 6
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27 | elpri 3627 |
. . . . . 6
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28 | 26, 27 | syl 14 |
. . . . 5
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29 | 28 | ad2antrr 488 |
. . . 4
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30 | 13, 19, 29 | mpjaodan 799 |
. . 3
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31 | nndceq0 4629 |
. . . . 5
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32 | exmiddc 837 |
. . . . 5
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33 | 31, 32 | syl 14 |
. . . 4
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34 | 33 | ad2antrr 488 |
. . 3
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35 | 5, 30, 34 | mpjaodan 799 |
. 2
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36 | 1n0 6446 |
. . . . . 6
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37 | 36 | neii 2359 |
. . . . 5
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38 | simpr 110 |
. . . . . . . 8
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39 | 38 | fveq1d 5529 |
. . . . . . 7
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40 | eqid 2187 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
41 | eleq1 2250 |
. . . . . . . . . . 11
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42 | 41 | ifbid 3567 |
. . . . . . . . . 10
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43 | id 19 |
. . . . . . . . . 10
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44 | nnord 4623 |
. . . . . . . . . . . . 13
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45 | ordirr 4553 |
. . . . . . . . . . . . 13
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46 | 44, 45 | syl 14 |
. . . . . . . . . . . 12
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47 | 46 | iffalsed 3556 |
. . . . . . . . . . 11
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48 | peano1 4605 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() | |
49 | 47, 48 | eqeltrdi 2278 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | 40, 42, 43, 49 | fvmptd3 5622 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
51 | 50, 47 | eqtrd 2220 |
. . . . . . . 8
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52 | 51 | ad3antrrr 492 |
. . . . . . 7
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53 | simplr 528 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
54 | 39, 52, 53 | 3eqtr3rd 2229 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
55 | 54 | ex 115 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
56 | 37, 55 | mtoi 665 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
57 | 56 | olcd 735 |
. . 3
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58 | df-dc 836 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
59 | 57, 58 | sylibr 134 |
. 2
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60 | simpl 109 |
. . . . 5
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61 | 21, 60 | ffvelcdmd 5665 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
62 | 61, 25 | eleqtrdi 2280 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
63 | elpri 3627 |
. . 3
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64 | 62, 63 | syl 14 |
. 2
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65 | 35, 59, 64 | mpjaodan 799 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1o 6430 df-2o 6431 df-map 6663 df-nninf 7132 |
This theorem is referenced by: (None) |
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