Step | Hyp | Ref
| Expression |
1 | | nns.s |
. 2
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ℕ∞               |
2 | | 1lt2o 6437 |
. . . . . . 7
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3 | 2 | a1i 9 |
. . . . . 6
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ℕ∞    |
4 | | nninff 7115 |
. . . . . . . 8
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5 | 4 | adantr 276 |
. . . . . . 7
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ℕ∞        |
6 | | nnpredcl 4619 |
. . . . . . . 8
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7 | 6 | adantl 277 |
. . . . . . 7
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ℕ∞     |
8 | 5, 7 | ffvelcdmd 5648 |
. . . . . 6
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ℕ∞         |
9 | | nndceq0 4614 |
. . . . . . 7
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DECID
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10 | 9 | adantl 277 |
. . . . . 6
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ℕ∞  DECID   |
11 | 3, 8, 10 | ifcldcd 3569 |
. . . . 5
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ℕ∞              |
12 | | eqid 2177 |
. . . . 5
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("        |
13 | 11, 12 | fmptd 5666 |
. . . 4
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14 | | 2onn 6516 |
. . . . 5
 |
15 | | omex 4589 |
. . . . 5
 |
16 | | elmapg 6655 |
. . . . 5
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("              |
17 | 14, 15, 16 | mp2an 426 |
. . . 4
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18 | 13, 17 | sylibr 134 |
. . 3
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)" |
19 | | 1on 6418 |
. . . . . . . . 9
 |
20 | 19 | ontrci 4424 |
. . . . . . . 8
 |
21 | 2 | a1i 9 |
. . . . . . . . . . 11
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ℕ∞    |
22 | 4 | adantr 276 |
. . . . . . . . . . . 12
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ℕ∞        |
23 | | peano2 4591 |
. . . . . . . . . . . . . 14
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  |
24 | 23 | adantl 277 |
. . . . . . . . . . . . 13
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ℕ∞ 
  |
25 | | nnpredcl 4619 |
. . . . . . . . . . . . 13
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26 | 24, 25 | syl 14 |
. . . . . . . . . . . 12
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ℕ∞     |
27 | 22, 26 | ffvelcdmd 5648 |
. . . . . . . . . . 11
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ℕ∞         |
28 | | nndceq0 4614 |
. . . . . . . . . . . 12
 DECID   |
29 | 24, 28 | syl 14 |
. . . . . . . . . . 11
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ℕ∞  DECID
  |
30 | 21, 27, 29 | ifcldcd 3569 |
. . . . . . . . . 10
 
ℕ∞              |
31 | 30 | adantr 276 |
. . . . . . . . 9
  
ℕ∞               |
32 | | df-2o 6412 |
. . . . . . . . 9
 |
33 | 31, 32 | eleqtrdi 2270 |
. . . . . . . 8
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ℕ∞               |
34 | | trsucss 4420 |
. . . . . . . 8
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                        |
35 | 20, 33, 34 | mpsyl 65 |
. . . . . . 7
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ℕ∞               |
36 | | iftrue 3539 |
. . . . . . . 8
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            |
37 | 36 | adantl 277 |
. . . . . . 7
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ℕ∞               |
38 | 35, 37 | sseqtrrd 3194 |
. . . . . 6
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ℕ∞                         |
39 | | simpr 110 |
. . . . . . . . . . . 12
 
ℕ∞    |
40 | 39 | adantr 276 |
. . . . . . . . . . 11
  
ℕ∞  
  |
41 | | nnord 4608 |
. . . . . . . . . . 11
   |
42 | | ordtr 4375 |
. . . . . . . . . . 11
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  |
43 | 40, 41, 42 | 3syl 17 |
. . . . . . . . . 10
  
ℕ∞  
  |
44 | | unisucg 4411 |
. . . . . . . . . . 11
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    |
45 | 40, 44 | syl 14 |
. . . . . . . . . 10
  
ℕ∞  
     |
46 | 43, 45 | mpbid 147 |
. . . . . . . . 9
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ℕ∞  

  |
47 | 46 | fveq2d 5515 |
. . . . . . . 8
  
ℕ∞  
           |
48 | | simpr 110 |
. . . . . . . . . . . 12
  
ℕ∞  
  |
49 | 48 | neqned 2354 |
. . . . . . . . . . 11
  
ℕ∞  
  |
50 | | nnsucpred 4613 |
. . . . . . . . . . 11
      |
51 | 40, 49, 50 | syl2anc 411 |
. . . . . . . . . 10
  
ℕ∞  

  |
52 | 51 | fveq2d 5515 |
. . . . . . . . 9
  
ℕ∞  
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        |
53 | | suceq 4399 |
. . . . . . . . . . . 12
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   |
54 | 53 | fveq2d 5515 |
. . . . . . . . . . 11
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("           |
55 | | fveq2 5511 |
. . . . . . . . . . 11
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("           |
56 | 54, 55 | sseq12d 3186 |
. . . . . . . . . 10
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("                      |
57 | | fveq1 5510 |
. . . . . . . . . . . . . . . 16
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f"         |
58 | | fveq1 5510 |
. . . . . . . . . . . . . . . 16
           |
59 | 57, 58 | sseq12d 3186 |
. . . . . . . . . . . . . . 15
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("           
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60 | 59 | ralbidv 2477 |
. . . . . . . . . . . . . 14
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61 | | df-nninf 7113 |
. . . . . . . . . . . . . 14
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62 | 60, 61 | elrab2 2896 |
. . . . . . . . . . . . 13
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63 | 62 | simprbi 275 |
. . . . . . . . . . . 12
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)" |
64 | | suceq 4399 |
. . . . . . . . . . . . . . 15
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65 | 64 | fveq2d 5515 |
. . . . . . . . . . . . . 14
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66 | | fveq2 5511 |
. . . . . . . . . . . . . 14
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67 | 65, 66 | sseq12d 3186 |
. . . . . . . . . . . . 13
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68 | 67 | cbvralv 2703 |
. . . . . . . . . . . 12
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p"     |
69 | 63, 68 | sylib 122 |
. . . . . . . . . . 11
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p"     |
70 | 69 | ad2antrr 488 |
. . . . . . . . . 10
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ℕ∞  
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p"     |
71 | | nnpredcl 4619 |
. . . . . . . . . . . 12
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72 | 71 | adantl 277 |
. . . . . . . . . . 11
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ℕ∞     |
73 | 72 | adantr 276 |
. . . . . . . . . 10
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ℕ∞  
   |
74 | 56, 70, 73 | rspcdva 2846 |
. . . . . . . . 9
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ℕ∞  
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       |
75 | 52, 74 | eqsstrrd 3192 |
. . . . . . . 8
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76 | 47, 75 | eqsstrd 3191 |
. . . . . . 7
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77 | | peano3 4592 |
. . . . . . . . . 10
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78 | 77 | neneqd 2368 |
. . . . . . . . 9
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  |
79 | 78 | ad2antlr 489 |
. . . . . . . 8
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ℕ∞  
  |
80 | 79 | iffalsed 3544 |
. . . . . . 7
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ℕ∞  
                 |
81 | 48 | iffalsed 3544 |
. . . . . . 7
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ℕ∞  
                 |
82 | 76, 80, 81 | 3sstr4d 3200 |
. . . . . 6
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ℕ∞  
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)"             |
83 | | nndceq0 4614 |
. . . . . . . 8
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DECID
  |
84 | 83 | adantl 277 |
. . . . . . 7
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ℕ∞  DECID   |
85 | | exmiddc 836 |
. . . . . . 7
DECID     |
86 | 84, 85 | syl 14 |
. . . . . 6
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ℕ∞      |
87 | 38, 82, 86 | mpjaodan 798 |
. . . . 5
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ℕ∞                        |
88 | | eqeq1 2184 |
. . . . . . . 8
 
   |
89 | | unieq 3816 |
. . . . . . . . 9
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90 | 89 | fveq2d 5515 |
. . . . . . . 8
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("            |
91 | 88, 90 | ifbieq2d 3558 |
. . . . . . 7
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)"   |
92 | 91, 12 | fvmptg 5588 |
. . . . . 6
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("                                   |
93 | 24, 30, 92 | syl2anc 411 |
. . . . 5
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ℕ∞                
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("        |
94 | 22, 72 | ffvelcdmd 5648 |
. . . . . . 7
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ℕ∞         |
95 | 21, 94, 84 | ifcldcd 3569 |
. . . . . 6
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ℕ∞              |
96 | | eqeq1 2184 |
. . . . . . . 8
 
   |
97 | | unieq 3816 |
. . . . . . . . 9
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98 | 97 | fveq2d 5515 |
. . . . . . . 8
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p"           |
99 | 96, 98 | ifbieq2d 3558 |
. . . . . . 7
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100 | 99, 12 | fvmptg 5588 |
. . . . . 6
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)"   |
101 | 39, 95, 100 | syl2anc 411 |
. . . . 5
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)"   |
102 | 87, 93, 101 | 3sstr4d 3200 |
. . . 4
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("            |
103 | 102 | ralrimiva 2550 |
. . 3
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("        
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("            |
104 | | fveq1 5510 |
. . . . . 6
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("                             |
105 | | fveq1 5510 |
. . . . . 6
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("                             |
106 | 104, 105 | sseq12d 3186 |
. . . . 5
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("          
                   
                    |
107 | 106 | ralbidv 2477 |
. . . 4
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108 | 107, 61 | elrab2 2896 |
. . 3
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109 | 18, 103, 108 | sylanbrc 417 |
. 2
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110 | 1, 109 | fmpti 5664 |
1
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