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Mirrors > Home > ILE Home > Th. List > nnnninf | Unicode version |
Description: Elements of ℕ∞ corresponding to natural numbers. The natural number corresponds to a sequence of ones followed by zeroes. Contrast to a sequence which is all ones as seen at infnninf 6990. Remark/TODO: the theorem still holds if , that is, the antecedent could be weakened to . (Contributed by Jim Kingdon, 14-Jul-2022.) |
Ref | Expression |
---|---|
nnnninf | ℕ∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1oex 6289 | . . . . . . . 8 | |
2 | 1 | sucid 4309 | . . . . . . 7 |
3 | df-2o 6282 | . . . . . . 7 | |
4 | 2, 3 | eleqtrri 2193 | . . . . . 6 |
5 | 4 | a1i 9 | . . . . 5 |
6 | 2on0 6291 | . . . . . . 7 | |
7 | 2onn 6385 | . . . . . . . 8 | |
8 | nn0eln0 4503 | . . . . . . . 8 | |
9 | 7, 8 | ax-mp 5 | . . . . . . 7 |
10 | 6, 9 | mpbir 145 | . . . . . 6 |
11 | 10 | a1i 9 | . . . . 5 |
12 | nndcel 6364 | . . . . . 6 DECID | |
13 | 12 | ancoms 266 | . . . . 5 DECID |
14 | 5, 11, 13 | ifcldcd 3477 | . . . 4 |
15 | eqid 2117 | . . . 4 | |
16 | 14, 15 | fmptd 5542 | . . 3 |
17 | 7 | elexi 2672 | . . . 4 |
18 | omex 4477 | . . . 4 | |
19 | 17, 18 | elmap 6539 | . . 3 |
20 | 16, 19 | sylibr 133 | . 2 |
21 | ssid 3087 | . . . . . . . . 9 | |
22 | iftrue 3449 | . . . . . . . . . . 11 | |
23 | 22 | sseq1d 3096 | . . . . . . . . . 10 |
24 | 23 | adantl 275 | . . . . . . . . 9 |
25 | 21, 24 | mpbiri 167 | . . . . . . . 8 |
26 | 0ss 3371 | . . . . . . . . 9 | |
27 | iffalse 3452 | . . . . . . . . . . 11 | |
28 | 27 | sseq1d 3096 | . . . . . . . . . 10 |
29 | 28 | adantl 275 | . . . . . . . . 9 |
30 | 26, 29 | mpbiri 167 | . . . . . . . 8 |
31 | peano2 4479 | . . . . . . . . . . 11 | |
32 | 31 | adantl 275 | . . . . . . . . . 10 |
33 | simpl 108 | . . . . . . . . . 10 | |
34 | nndcel 6364 | . . . . . . . . . 10 DECID | |
35 | 32, 33, 34 | syl2anc 408 | . . . . . . . . 9 DECID |
36 | exmiddc 806 | . . . . . . . . 9 DECID | |
37 | 35, 36 | syl 14 | . . . . . . . 8 |
38 | 25, 30, 37 | mpjaodan 772 | . . . . . . 7 |
39 | 38 | adantr 274 | . . . . . 6 |
40 | iftrue 3449 | . . . . . . 7 | |
41 | 40 | adantl 275 | . . . . . 6 |
42 | 39, 41 | sseqtrrd 3106 | . . . . 5 |
43 | ssid 3087 | . . . . . . 7 | |
44 | 43 | a1i 9 | . . . . . 6 |
45 | nnord 4495 | . . . . . . . . . . . 12 | |
46 | ordtr 4270 | . . . . . . . . . . . 12 | |
47 | 45, 46 | syl 14 | . . . . . . . . . . 11 |
48 | trsuc 4314 | . . . . . . . . . . 11 | |
49 | 47, 48 | sylan 281 | . . . . . . . . . 10 |
50 | 49 | ex 114 | . . . . . . . . 9 |
51 | 50 | adantr 274 | . . . . . . . 8 |
52 | 51 | con3dimp 609 | . . . . . . 7 |
53 | 52, 27 | syl 14 | . . . . . 6 |
54 | iffalse 3452 | . . . . . . 7 | |
55 | 54 | adantl 275 | . . . . . 6 |
56 | 44, 53, 55 | 3sstr4d 3112 | . . . . 5 |
57 | nndcel 6364 | . . . . . . 7 DECID | |
58 | 57 | ancoms 266 | . . . . . 6 DECID |
59 | exmiddc 806 | . . . . . 6 DECID | |
60 | 58, 59 | syl 14 | . . . . 5 |
61 | 42, 56, 60 | mpjaodan 772 | . . . 4 |
62 | 4 | a1i 9 | . . . . . 6 |
63 | 10 | a1i 9 | . . . . . 6 |
64 | 62, 63, 35 | ifcldcd 3477 | . . . . 5 |
65 | eleq1 2180 | . . . . . . 7 | |
66 | 65 | ifbid 3463 | . . . . . 6 |
67 | 66, 15 | fvmptg 5465 | . . . . 5 |
68 | 32, 64, 67 | syl2anc 408 | . . . 4 |
69 | simpr 109 | . . . . 5 | |
70 | 62, 63, 58 | ifcldcd 3477 | . . . . 5 |
71 | eleq1 2180 | . . . . . . 7 | |
72 | 71 | ifbid 3463 | . . . . . 6 |
73 | 72, 15 | fvmptg 5465 | . . . . 5 |
74 | 69, 70, 73 | syl2anc 408 | . . . 4 |
75 | 61, 68, 74 | 3sstr4d 3112 | . . 3 |
76 | 75 | ralrimiva 2482 | . 2 |
77 | fveq1 5388 | . . . . 5 | |
78 | fveq1 5388 | . . . . 5 | |
79 | 77, 78 | sseq12d 3098 | . . . 4 |
80 | 79 | ralbidv 2414 | . . 3 |
81 | df-nninf 6975 | . . 3 ℕ∞ | |
82 | 80, 81 | elrab2 2816 | . 2 ℕ∞ |
83 | 20, 76, 82 | sylanbrc 413 | 1 ℕ∞ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 DECID wdc 804 wceq 1316 wcel 1465 wne 2285 wral 2393 wss 3041 c0 3333 cif 3444 cmpt 3959 wtr 3996 word 4254 csuc 4257 com 4474 wf 5089 cfv 5093 (class class class)co 5742 c1o 6274 c2o 6275 cmap 6510 ℕ∞xnninf 6973 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1o 6281 df-2o 6282 df-map 6512 df-nninf 6975 |
This theorem is referenced by: fnn0nninf 10165 nninfsellemdc 13102 nninfsellemqall 13107 nninffeq 13112 |
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