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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 109 |
. . . . . . 7
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3 | 2 | renegcld 8009 |
. . . . . 6
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4 | simpr 109 |
. . . . . . 7
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5 | 4 | renegcld 8009 |
. . . . . 6
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6 | recn 7625 |
. . . . . . . 8
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7 | recn 7625 |
. . . . . . . 8
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8 | negcon2 7886 |
. . . . . . . 8
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9 | 6, 7, 8 | syl2an 285 |
. . . . . . 7
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10 | 9 | adantl 273 |
. . . . . 6
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11 | 1, 3, 5, 10 | f1ocnv2d 5906 |
. . . . 5
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12 | 11 | mptru 1308 |
. . . 4
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13 | 12 | simpli 110 |
. . 3
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14 | simpl 108 |
. . . . . . . 8
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15 | 14 | recnd 7666 |
. . . . . . 7
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16 | 15 | negcld 7931 |
. . . . . 6
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17 | 7 | adantl 273 |
. . . . . . 7
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18 | 17 | negcld 7931 |
. . . . . 6
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19 | brcnvg 4658 |
. . . . . 6
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20 | 16, 18, 19 | syl2anc 406 |
. . . . 5
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21 | 1 | a1i 9 |
. . . . . . 7
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22 | negeq 7826 |
. . . . . . . 8
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23 | 22 | adantl 273 |
. . . . . . 7
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24 | 21, 23, 14, 16 | fvmptd 5434 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | negeq 7826 |
. . . . . . . 8
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26 | 25 | adantl 273 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | simpr 109 |
. . . . . . 7
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28 | 21, 26, 27, 18 | fvmptd 5434 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 24, 28 | breq12d 3888 |
. . . . 5
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30 | ltneg 8091 |
. . . . 5
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31 | 20, 29, 30 | 3bitr4rd 220 |
. . . 4
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32 | 31 | rgen2a 2445 |
. . 3
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33 | df-isom 5068 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
34 | 13, 32, 33 | mpbir2an 894 |
. 2
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35 | negeq 7826 |
. . . 4
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36 | 35 | cbvmptv 3964 |
. . 3
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37 | 12 | simpri 112 |
. . 3
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38 | 36, 37, 1 | 3eqtr4i 2130 |
. 2
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39 | 34, 38 | pm3.2i 268 |
1
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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 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-addcom 7595 ax-addass 7597 ax-distr 7599 ax-i2m1 7600 ax-0id 7603 ax-rnegex 7604 ax-cnre 7606 ax-pre-ltadd 7611 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-isom 5068 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-pnf 7674 df-mnf 7675 df-ltxr 7677 df-sub 7806 df-neg 7807 |
This theorem is referenced by: infrenegsupex 9239 |
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