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Mirrors > Home > ILE Home > Th. List > negiso | Unicode version |
Description: Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
negiso.1 |
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Ref | Expression |
---|---|
negiso |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negiso.1 |
. . . . . 6
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2 | simpr 110 |
. . . . . . 7
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3 | 2 | renegcld 8362 |
. . . . . 6
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4 | simpr 110 |
. . . . . . 7
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5 | 4 | renegcld 8362 |
. . . . . 6
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6 | recn 7969 |
. . . . . . . 8
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7 | recn 7969 |
. . . . . . . 8
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8 | negcon2 8235 |
. . . . . . . 8
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9 | 6, 7, 8 | syl2an 289 |
. . . . . . 7
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10 | 9 | adantl 277 |
. . . . . 6
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11 | 1, 3, 5, 10 | f1ocnv2d 6094 |
. . . . 5
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12 | 11 | mptru 1373 |
. . . 4
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13 | 12 | simpli 111 |
. . 3
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14 | simpl 109 |
. . . . . . . 8
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15 | 14 | recnd 8011 |
. . . . . . 7
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16 | 15 | negcld 8280 |
. . . . . 6
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17 | 7 | adantl 277 |
. . . . . . 7
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18 | 17 | negcld 8280 |
. . . . . 6
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19 | brcnvg 4823 |
. . . . . 6
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20 | 16, 18, 19 | syl2anc 411 |
. . . . 5
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21 | 1 | a1i 9 |
. . . . . . 7
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22 | negeq 8175 |
. . . . . . . 8
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23 | 22 | adantl 277 |
. . . . . . 7
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24 | 21, 23, 14, 16 | fvmptd 5614 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | negeq 8175 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | 25 | adantl 277 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | simpr 110 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 21, 26, 27, 18 | fvmptd 5614 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 24, 28 | breq12d 4031 |
. . . . 5
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30 | ltneg 8444 |
. . . . 5
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31 | 20, 29, 30 | 3bitr4rd 221 |
. . . 4
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32 | 31 | rgen2a 2544 |
. . 3
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33 | df-isom 5241 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
34 | 13, 32, 33 | mpbir2an 944 |
. 2
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35 | negeq 8175 |
. . . 4
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36 | 35 | cbvmptv 4114 |
. . 3
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37 | 12 | simpri 113 |
. . 3
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38 | 36, 37, 1 | 3eqtr4i 2220 |
. 2
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39 | 34, 38 | pm3.2i 272 |
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-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-addcom 7936 ax-addass 7938 ax-distr 7940 ax-i2m1 7941 ax-0id 7944 ax-rnegex 7945 ax-cnre 7947 ax-pre-ltadd 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-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 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 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-isom 5241 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-pnf 8019 df-mnf 8020 df-ltxr 8022 df-sub 8155 df-neg 8156 |
This theorem is referenced by: infrenegsupex 9619 |
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