Step | Hyp | Ref
| Expression |
1 | | peano4nninf.s |
. . 3
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2 | 1 | nnsf 14410 |
. 2
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3 | | fveq1 5510 |
. . . . . . . . . . 11
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4 | | fveq1 5510 |
. . . . . . . . . . 11
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5 | 3, 4 | sseq12d 3186 |
. . . . . . . . . 10
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6 | 5 | ralbidv 2477 |
. . . . . . . . 9
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7 | | df-nninf 7113 |
. . . . . . . . 9
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8 | 6, 7 | elrab2 2896 |
. . . . . . . 8
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9 | 8 | simplbi 274 |
. . . . . . 7
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10 | | elmapfn 6665 |
. . . . . . 7
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11 | 9, 10 | syl 14 |
. . . . . 6
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12 | 11 | ad2antrr 488 |
. . . . 5
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13 | | fveq1 5510 |
. . . . . . . . . . 11
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14 | | fveq1 5510 |
. . . . . . . . . . 11
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15 | 13, 14 | sseq12d 3186 |
. . . . . . . . . 10
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f"  alt="`
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       |
16 | 15 | ralbidv 2477 |
. . . . . . . . 9
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f"           |
17 | 16, 7 | elrab2 2896 |
. . . . . . . 8
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y"          |
18 | 17 | simplbi 274 |
. . . . . . 7
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19 | | elmapfn 6665 |
. . . . . . 7
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  |
20 | 18, 19 | syl 14 |
. . . . . 6
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21 | 20 | ad2antlr 489 |
. . . . 5
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ℕ∞
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)"   |
22 | | simplr 528 |
. . . . . . 7
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23 | 22 | fveq1d 5513 |
. . . . . 6
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24 | | fveq1 5510 |
. . . . . . . . . . . 12
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`"          |
25 | 24 | ifeq2d 3552 |
. . . . . . . . . . 11
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26 | 25 | mpteq2dv 4091 |
. . . . . . . . . 10
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27 | | omex 4589 |
. . . . . . . . . . 11
 |
28 | 27 | mptex 5738 |
. . . . . . . . . 10
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29 | 26, 1, 28 | fvmpt 5589 |
. . . . . . . . 9
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30 | 29 | ad3antrrr 492 |
. . . . . . . 8
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ℕ∞                  |
31 | | simpr 110 |
. . . . . . . . . 10
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)"     |
32 | 31 | eqeq1d 2186 |
. . . . . . . . 9
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ℕ∞            
   |
33 | 31 | unieqd 3818 |
. . . . . . . . . 10
     ℕ∞
ℕ∞            
   |
34 | 33 | fveq2d 5515 |
. . . . . . . . 9
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35 | 32, 34 | ifbieq2d 3558 |
. . . . . . . 8
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36 | | peano2 4591 |
. . . . . . . . 9
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37 | 36 | adantl 277 |
. . . . . . . 8
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38 | | 1lt2o 6437 |
. . . . . . . . . 10
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39 | 38 | a1i 9 |
. . . . . . . . 9
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40 | | nninff 7115 |
. . . . . . . . . . 11
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41 | 40 | ad3antrrr 492 |
. . . . . . . . . 10
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42 | | nnpredcl 4619 |
. . . . . . . . . . 11
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43 | 37, 42 | syl 14 |
. . . . . . . . . 10
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44 | 41, 43 | ffvelcdmd 5648 |
. . . . . . . . 9
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45 | | nndceq0 4614 |
. . . . . . . . . 10
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46 | 37, 45 | syl 14 |
. . . . . . . . 9
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47 | 39, 44, 46 | ifcldcd 3569 |
. . . . . . . 8
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48 | 30, 35, 37, 47 | fvmptd 5593 |
. . . . . . 7
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`"              |
49 | | peano3 4592 |
. . . . . . . . . 10
   |
50 | 49 | adantl 277 |
. . . . . . . . 9
    ℕ∞
ℕ∞             |
51 | 50 | neneqd 2368 |
. . . . . . . 8
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ℕ∞          
  |
52 | 51 | iffalsed 3544 |
. . . . . . 7
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ℕ∞                            |
53 | | nnord 4608 |
. . . . . . . . . . 11
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54 | | ordtr 4375 |
. . . . . . . . . . 11
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  |
55 | 53, 54 | syl 14 |
. . . . . . . . . 10
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56 | | unisucg 4411 |
. . . . . . . . . 10
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57 | 55, 56 | mpbid 147 |
. . . . . . . . 9
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58 | 57 | fveq2d 5515 |
. . . . . . . 8
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59 | 58 | adantl 277 |
. . . . . . 7
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60 | 48, 52, 59 | 3eqtrd 2214 |
. . . . . 6
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61 | | fveq1 5510 |
. . . . . . . . . . . 12
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62 | 61 | ifeq2d 3552 |
. . . . . . . . . . 11
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63 | 62 | mpteq2dv 4091 |
. . . . . . . . . 10
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64 | 27 | mptex 5738 |
. . . . . . . . . 10
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65 | 63, 1, 64 | fvmpt 5589 |
. . . . . . . . 9
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66 | 65 | ad3antlr 493 |
. . . . . . . 8
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)"                    |
67 | 33 | fveq2d 5515 |
. . . . . . . . 9
     ℕ∞
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)"               |
68 | 32, 67 | ifbieq2d 3558 |
. . . . . . . 8
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)"   |
69 | | nninff 7115 |
. . . . . . . . . . 11
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70 | 69 | ad3antlr 493 |
. . . . . . . . . 10
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)"        |
71 | 70, 43 | ffvelcdmd 5648 |
. . . . . . . . 9
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ℕ∞                  |
72 | 39, 71, 46 | ifcldcd 3569 |
. . . . . . . 8
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ℕ∞                       |
73 | 66, 68, 37, 72 | fvmptd 5593 |
. . . . . . 7
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74 | 51 | iffalsed 3544 |
. . . . . . 7
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75 | 57 | fveq2d 5515 |
. . . . . . . 8
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76 | 75 | adantl 277 |
. . . . . . 7
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77 | 73, 74, 76 | 3eqtrd 2214 |
. . . . . 6
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78 | 23, 60, 77 | 3eqtr3d 2218 |
. . . . 5
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ℕ∞                    |
79 | 12, 21, 78 | eqfnfvd 5612 |
. . . 4
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ℕ∞
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80 | 79 | ex 115 |
. . 3
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ℕ∞
ℕ∞
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81 | 80 | rgen2a 2531 |
. 2
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82 | | dff13 5763 |
. 2
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83 | 2, 81, 82 | mpbir2an 942 |
1
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