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Mirrors > Home > ILE Home > Th. List > negf1o | Unicode version |
Description: Negation is an isomorphism of a subset of the real numbers to the negated elements of the subset. (Contributed by AV, 9-Aug-2020.) |
Ref | Expression |
---|---|
negf1o.1 |
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Ref | Expression |
---|---|
negf1o |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negf1o.1 |
. . 3
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2 | ssel 3055 |
. . . . . 6
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3 | renegcl 7939 |
. . . . . 6
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4 | 2, 3 | syl6 33 |
. . . . 5
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5 | 4 | imp 123 |
. . . 4
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6 | 2 | imp 123 |
. . . . 5
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7 | recn 7670 |
. . . . . . . . 9
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8 | negneg 7928 |
. . . . . . . . . 10
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9 | 8 | eqcomd 2118 |
. . . . . . . . 9
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10 | 7, 9 | syl 14 |
. . . . . . . 8
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11 | 10 | eleq1d 2181 |
. . . . . . 7
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12 | 11 | biimpcd 158 |
. . . . . 6
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13 | 12 | adantl 273 |
. . . . 5
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14 | 6, 13 | mpd 13 |
. . . 4
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15 | negeq 7871 |
. . . . . 6
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16 | 15 | eleq1d 2181 |
. . . . 5
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17 | 16 | elrab 2807 |
. . . 4
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18 | 5, 14, 17 | sylanbrc 411 |
. . 3
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19 | negeq 7871 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 19 | eleq1d 2181 |
. . . . . 6
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21 | 20 | elrab 2807 |
. . . . 5
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22 | simpr 109 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 22 | a1i 9 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 21, 23 | syl5bi 151 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | imp 123 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 2, 7 | syl6com 35 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 26 | adantl 273 |
. . . . . . . . 9
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28 | 27 | imp 123 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | recn 7670 |
. . . . . . . . 9
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30 | 29 | ad3antrrr 481 |
. . . . . . . 8
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31 | negcon2 7931 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
32 | 28, 30, 31 | syl2anc 406 |
. . . . . . 7
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33 | 32 | exp31 359 |
. . . . . 6
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34 | 21, 33 | sylbi 120 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 34 | impcom 124 |
. . . 4
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36 | 35 | impcom 124 |
. . 3
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37 | 1, 18, 25, 36 | f1ocnv2d 5926 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | 37 | simpld 111 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-setind 4410 ax-resscn 7630 ax-1cn 7631 ax-icn 7633 ax-addcl 7634 ax-addrcl 7635 ax-mulcl 7636 ax-addcom 7638 ax-addass 7640 ax-distr 7642 ax-i2m1 7643 ax-0id 7646 ax-rnegex 7647 ax-cnre 7649 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-sub 7851 df-neg 7852 |
This theorem is referenced by: negfi 10884 |
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