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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 8336 |
. . . . . 6
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4 | simpr 110 |
. . . . . . 7
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5 | 4 | renegcld 8336 |
. . . . . 6
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6 | recn 7943 |
. . . . . . . 8
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7 | recn 7943 |
. . . . . . . 8
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8 | negcon2 8209 |
. . . . . . . 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 6074 |
. . . . 5
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12 | 11 | mptru 1362 |
. . . 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 7985 |
. . . . . . 7
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16 | 15 | negcld 8254 |
. . . . . 6
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17 | 7 | adantl 277 |
. . . . . . 7
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18 | 17 | negcld 8254 |
. . . . . 6
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19 | brcnvg 4808 |
. . . . . 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 8149 |
. . . . . . . 8
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23 | 22 | adantl 277 |
. . . . . . 7
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24 | 21, 23, 14, 16 | fvmptd 5597 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | negeq 8149 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | 25 | adantl 277 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | simpr 110 |
. . . . . . 7
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28 | 21, 26, 27, 18 | fvmptd 5597 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 24, 28 | breq12d 4016 |
. . . . 5
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30 | ltneg 8418 |
. . . . 5
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31 | 20, 29, 30 | 3bitr4rd 221 |
. . . 4
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32 | 31 | rgen2a 2531 |
. . 3
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33 | df-isom 5225 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
34 | 13, 32, 33 | mpbir2an 942 |
. 2
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35 | negeq 8149 |
. . . 4
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36 | 35 | cbvmptv 4099 |
. . 3
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37 | 12 | simpri 113 |
. . 3
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38 | 36, 37, 1 | 3eqtr4i 2208 |
. 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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7993 df-mnf 7994 df-ltxr 7996 df-sub 8129 df-neg 8130 |
This theorem is referenced by: infrenegsupex 9593 |
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