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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 |
Ref | Expression |
---|---|
negf1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negf1o.1 | . . 3 | |
2 | ssel 3061 | . . . . . 6 | |
3 | renegcl 7991 | . . . . . 6 | |
4 | 2, 3 | syl6 33 | . . . . 5 |
5 | 4 | imp 123 | . . . 4 |
6 | 2 | imp 123 | . . . . 5 |
7 | recn 7721 | . . . . . . . . 9 | |
8 | negneg 7980 | . . . . . . . . . 10 | |
9 | 8 | eqcomd 2123 | . . . . . . . . 9 |
10 | 7, 9 | syl 14 | . . . . . . . 8 |
11 | 10 | eleq1d 2186 | . . . . . . 7 |
12 | 11 | biimpcd 158 | . . . . . 6 |
13 | 12 | adantl 275 | . . . . 5 |
14 | 6, 13 | mpd 13 | . . . 4 |
15 | negeq 7923 | . . . . . 6 | |
16 | 15 | eleq1d 2186 | . . . . 5 |
17 | 16 | elrab 2813 | . . . 4 |
18 | 5, 14, 17 | sylanbrc 413 | . . 3 |
19 | negeq 7923 | . . . . . . 7 | |
20 | 19 | eleq1d 2186 | . . . . . 6 |
21 | 20 | elrab 2813 | . . . . 5 |
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 275 | . . . . . . . . 9 |
28 | 27 | imp 123 | . . . . . . . 8 |
29 | recn 7721 | . . . . . . . . 9 | |
30 | 29 | ad3antrrr 483 | . . . . . . . 8 |
31 | negcon2 7983 | . . . . . . . 8 | |
32 | 28, 30, 31 | syl2anc 408 | . . . . . . 7 |
33 | 32 | exp31 361 | . . . . . 6 |
34 | 21, 33 | sylbi 120 | . . . . 5 |
35 | 34 | impcom 124 | . . . 4 |
36 | 35 | impcom 124 | . . 3 |
37 | 1, 18, 25, 36 | f1ocnv2d 5942 | . 2 |
38 | 37 | simpld 111 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 crab 2397 wss 3041 cmpt 3959 ccnv 4508 wf1o 5092 cc 7586 cr 7587 cneg 7902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 df-neg 7904 |
This theorem is referenced by: negfi 10967 |
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