Step | Hyp | Ref
| Expression |
1 | | txdis1cn.f |
. . 3
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2 | | txdis1cn.j |
. . . . . . 7
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3 | 2 | adantr 276 |
. . . . . 6
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4 | | txdis1cn.k |
. . . . . . . 8
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5 | | toptopon2 13184 |
. . . . . . . 8
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6 | 4, 5 | sylib 122 |
. . . . . . 7
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7 | 6 | adantr 276 |
. . . . . 6
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8 | | txdis1cn.1 |
. . . . . 6
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9 | | cnf2 13372 |
. . . . . 6
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10 | 3, 7, 8, 9 | syl3anc 1238 |
. . . . 5
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11 | | eqid 2177 |
. . . . . 6
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12 | 11 | fmpt 5662 |
. . . . 5
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)" |
13 | 10, 12 | sylibr 134 |
. . . 4
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U."   |
14 | 13 | ralrimiva 2550 |
. . 3
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)" |
15 | | ffnov 5973 |
. . 3
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16 | 1, 14, 15 | sylanbrc 417 |
. 2
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17 | | cnvimass 4987 |
. . . . . . . 8
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18 | 1 | adantr 276 |
. . . . . . . . 9
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)"  |
19 | | fndm 5311 |
. . . . . . . . 9
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20 | 18, 19 | syl 14 |
. . . . . . . 8
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21 | 17, 20 | sseqtrid 3205 |
. . . . . . 7
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22 | | relxp 4732 |
. . . . . . 7
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23 | | relss 4710 |
. . . . . . 7
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24 | 21, 22, 23 | mpisyl 1446 |
. . . . . 6
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        |
25 | | elpreima 5631 |
. . . . . . . 8
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u"
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26 | 18, 25 | syl 14 |
. . . . . . 7
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u"
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27 | | opelxp 4653 |
. . . . . . . . 9
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x"        |
28 | | df-ov 5872 |
. . . . . . . . . . 11
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29 | 28 | eqcomi 2181 |
. . . . . . . . . 10
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>."
     |
30 | 29 | eleq1i 2243 |
. . . . . . . . 9
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31 | 27, 30 | anbi12i 460 |
. . . . . . . 8
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32 | | simprll 537 |
. . . . . . . . . . . 12
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33 | | snelpwi 4209 |
. . . . . . . . . . . 12
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34 | 32, 33 | syl 14 |
. . . . . . . . . . 11
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35 | 11 | mptpreima 5118 |
. . . . . . . . . . . 12
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36 | 8 | adantrr 479 |
. . . . . . . . . . . . . 14
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37 | 36 | ad2ant2r 509 |
. . . . . . . . . . . . 13
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38 | | simplr 528 |
. . . . . . . . . . . . 13
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39 | | cnima 13387 |
. . . . . . . . . . . . 13
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40 | 37, 38, 39 | syl2anc 411 |
. . . . . . . . . . . 12
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("             |
41 | 35, 40 | eqeltrrid 2265 |
. . . . . . . . . . 11
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42 | | simprlr 538 |
. . . . . . . . . . . 12
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43 | | simprr 531 |
. . . . . . . . . . . 12
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44 | | vsnid 3623 |
. . . . . . . . . . . . . 14
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45 | | opelxp 4653 |
. . . . . . . . . . . . . 14
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46 | 44, 45 | mpbiran 940 |
. . . . . . . . . . . . 13
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47 | | oveq2 5877 |
. . . . . . . . . . . . . . 15
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48 | 47 | eleq1d 2246 |
. . . . . . . . . . . . . 14
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49 | 48 | elrab 2893 |
. . . . . . . . . . . . 13
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x"      |
50 | 46, 49 | bitri 184 |
. . . . . . . . . . . 12
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x"      |
51 | 42, 43, 50 | sylanbrc 417 |
. . . . . . . . . . 11
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52 | | relxp 4732 |
. . . . . . . . . . . . 13
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53 | 52 | a1i 9 |
. . . . . . . . . . . 12
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54 | | opelxp 4653 |
. . . . . . . . . . . . 13
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x"       |
55 | 32 | snssd 3736 |
. . . . . . . . . . . . . . . . . 18
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56 | 55 | sselda 3155 |
. . . . . . . . . . . . . . . . 17
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x"       
  |
57 | 56 | adantrr 479 |
. . . . . . . . . . . . . . . 16
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58 | | elrabi 2890 |
. . . . . . . . . . . . . . . . 17
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59 | 58 | ad2antll 491 |
. . . . . . . . . . . . . . . 16
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60 | 57, 59 | opelxpd 4656 |
. . . . . . . . . . . . . . 15
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61 | | df-ov 5872 |
. . . . . . . . . . . . . . . . 17
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62 | | elsni 3609 |
. . . . . . . . . . . . . . . . . . 19
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63 | 62 | ad2antrl 490 |
. . . . . . . . . . . . . . . . . 18
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64 | 63 | oveq1d 5884 |
. . . . . . . . . . . . . . . . 17
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65 | 61, 64 | eqtr3id 2224 |
. . . . . . . . . . . . . . . 16
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66 | | oveq2 5877 |
. . . . . . . . . . . . . . . . . . . 20
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67 | 66 | eleq1d 2246 |
. . . . . . . . . . . . . . . . . . 19
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68 | 67 | elrab 2893 |
. . . . . . . . . . . . . . . . . 18
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69 | 68 | simprbi 275 |
. . . . . . . . . . . . . . . . 17
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x"     |
70 | 69 | ad2antll 491 |
. . . . . . . . . . . . . . . 16
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x"     
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71 | 65, 70 | eqeltrd 2254 |
. . . . . . . . . . . . . . 15
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72 | | elpreima 5631 |
. . . . . . . . . . . . . . . . 17
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u"
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73 | 1, 72 | syl 14 |
. . . . . . . . . . . . . . . 16
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74 | 73 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
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75 | 60, 71, 74 | mpbir2and 944 |
. . . . . . . . . . . . . 14
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76 | 75 | ex 115 |
. . . . . . . . . . . . 13
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77 | 54, 76 | biimtrid 152 |
. . . . . . . . . . . 12
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u"    |
78 | 53, 77 | relssdv 4715 |
. . . . . . . . . . 11
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79 | | xpeq1 4637 |
. . . . . . . . . . . . . 14
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)"  |
80 | 79 | eleq2d 2247 |
. . . . . . . . . . . . 13
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81 | 79 | sseq1d 3184 |
. . . . . . . . . . . . 13
                       |
82 | 80, 81 | anbi12d 473 |
. . . . . . . . . . . 12
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83 | | xpeq2 4638 |
. . . . . . . . . . . . . 14
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84 | 83 | eleq2d 2247 |
. . . . . . . . . . . . 13
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85 | 83 | sseq1d 3184 |
. . . . . . . . . . . . 13
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86 | 84, 85 | anbi12d 473 |
. . . . . . . . . . . 12
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          |
87 | 82, 86 | rspc2ev 2856 |
. . . . . . . . . . 11
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88 | 34, 41, 51, 78, 87 | syl112anc 1242 |
. . . . . . . . . 10
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89 | | vex 2740 |
. . . . . . . . . . . 12
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_V" |
90 | | vex 2740 |
. . . . . . . . . . . 12
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91 | 89, 90 | opex 4226 |
. . . . . . . . . . 11
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92 | | eleq1 2240 |
. . . . . . . . . . . . 13
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93 | 92 | anbi1d 465 |
. . . . . . . . . . . 12
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94 | 93 | 2rexbidv 2502 |
. . . . . . . . . . 11
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95 | 91, 94 | elab 2881 |
. . . . . . . . . 10
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96 | 88, 95 | sylibr 134 |
. . . . . . . . 9
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97 | 96 | ex 115 |
. . . . . . . 8
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98 | 31, 97 | biimtrid 152 |
. . . . . . 7
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99 | 26, 98 | sylbid 150 |
. . . . . 6
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100 | 24, 99 | relssdv 4715 |
. . . . 5
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101 | | ssabral 3226 |
. . . . 5
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u"    
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102 | 100, 101 | sylib 122 |
. . . 4
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u"    
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103 | | txdis1cn.x |
. . . . . . 7
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104 | | distopon 13254 |
. . . . . . 7
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TopOn    |
105 | 103, 104 | syl 14 |
. . . . . 6
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106 | 105 | adantr 276 |
. . . . 5
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107 | 2 | adantr 276 |
. . . . 5
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108 | | eltx 13426 |
. . . . 5
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109 | 106, 107,
108 | syl2anc 411 |
. . . 4
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110 | 102, 109 | mpbird 167 |
. . 3
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111 | 110 | ralrimiva 2550 |
. 2
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112 | | txtopon 13429 |
. . . 4
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113 | 105, 2, 112 | syl2anc 411 |
. . 3
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114 | | iscn 13364 |
. . 3
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115 | 113, 6, 114 | syl2anc 411 |
. 2
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116 | 16, 111, 115 | mpbir2and 944 |
1
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