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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 3164 |
. . . . . 6
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3 | renegcl 8248 |
. . . . . 6
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4 | 2, 3 | syl6 33 |
. . . . 5
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5 | 4 | imp 124 |
. . . 4
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6 | 2 | imp 124 |
. . . . 5
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7 | recn 7974 |
. . . . . . . . 9
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8 | negneg 8237 |
. . . . . . . . . 10
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9 | 8 | eqcomd 2195 |
. . . . . . . . 9
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10 | 7, 9 | syl 14 |
. . . . . . . 8
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11 | 10 | eleq1d 2258 |
. . . . . . 7
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12 | 11 | biimpcd 159 |
. . . . . 6
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13 | 12 | adantl 277 |
. . . . 5
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14 | 6, 13 | mpd 13 |
. . . 4
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15 | negeq 8180 |
. . . . . 6
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16 | 15 | eleq1d 2258 |
. . . . 5
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17 | 16 | elrab 2908 |
. . . 4
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18 | 5, 14, 17 | sylanbrc 417 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | negeq 8180 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 19 | eleq1d 2258 |
. . . . . 6
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21 | 20 | elrab 2908 |
. . . . 5
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22 | simpr 110 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 22 | a1i 9 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 21, 23 | biimtrid 152 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | imp 124 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 2, 7 | syl6com 35 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 26 | adantl 277 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 27 | imp 124 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | recn 7974 |
. . . . . . . . 9
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30 | 29 | ad3antrrr 492 |
. . . . . . . 8
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31 | negcon2 8240 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
32 | 28, 30, 31 | syl2anc 411 |
. . . . . . 7
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33 | 32 | exp31 364 |
. . . . . 6
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34 | 21, 33 | sylbi 121 |
. . . . 5
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35 | 34 | impcom 125 |
. . . 4
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36 | 35 | impcom 125 |
. . 3
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37 | 1, 18, 25, 36 | f1ocnv2d 6098 |
. 2
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38 | 37 | simpld 112 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-resscn 7933 ax-1cn 7934 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-distr 7945 ax-i2m1 7946 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-sub 8160 df-neg 8161 |
This theorem is referenced by: negfi 11268 |
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