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Mirrors > Home > ILE Home > Th. List > negfi | Unicode version |
Description: The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.) |
Ref | Expression |
---|---|
negfi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 2993 | . . . . . . . . . 10 | |
2 | renegcl 7369 | . . . . . . . . . 10 | |
3 | 1, 2 | syl6 33 | . . . . . . . . 9 |
4 | 3 | imp 122 | . . . . . . . 8 |
5 | 4 | ralrimiva 2434 | . . . . . . 7 |
6 | dmmptg 4838 | . . . . . . 7 | |
7 | 5, 6 | syl 14 | . . . . . 6 |
8 | 7 | eqcomd 2086 | . . . . 5 |
9 | 8 | eleq1d 2147 | . . . 4 |
10 | funmpt 4958 | . . . . 5 | |
11 | fundmfibi 6390 | . . . . 5 | |
12 | 10, 11 | mp1i 10 | . . . 4 |
13 | 9, 12 | bitr4d 189 | . . 3 |
14 | reex 7107 | . . . . . 6 | |
15 | 14 | ssex 3915 | . . . . 5 |
16 | mptexg 5407 | . . . . 5 | |
17 | 15, 16 | syl 14 | . . . 4 |
18 | eqid 2081 | . . . . . 6 | |
19 | 18 | negf1o 7486 | . . . . 5 |
20 | f1of1 5145 | . . . . 5 | |
21 | 19, 20 | syl 14 | . . . 4 |
22 | f1vrnfibi 6394 | . . . 4 | |
23 | 17, 21, 22 | syl2anc 403 | . . 3 |
24 | 1 | imp 122 | . . . . . . . . . 10 |
25 | 2 | adantl 271 | . . . . . . . . . . 11 |
26 | recn 7106 | . . . . . . . . . . . . . . . . 17 | |
27 | 26 | negnegd 7410 | . . . . . . . . . . . . . . . 16 |
28 | 27 | eqcomd 2086 | . . . . . . . . . . . . . . 15 |
29 | 28 | eleq1d 2147 | . . . . . . . . . . . . . 14 |
30 | 29 | biimpcd 157 | . . . . . . . . . . . . 13 |
31 | 30 | adantl 271 | . . . . . . . . . . . 12 |
32 | 31 | imp 122 | . . . . . . . . . . 11 |
33 | 25, 32 | jca 300 | . . . . . . . . . 10 |
34 | 24, 33 | mpdan 412 | . . . . . . . . 9 |
35 | eleq1 2141 | . . . . . . . . . 10 | |
36 | negeq 7301 | . . . . . . . . . . 11 | |
37 | 36 | eleq1d 2147 | . . . . . . . . . 10 |
38 | 35, 37 | anbi12d 456 | . . . . . . . . 9 |
39 | 34, 38 | syl5ibrcom 155 | . . . . . . . 8 |
40 | 39 | rexlimdva 2477 | . . . . . . 7 |
41 | simprr 498 | . . . . . . . . 9 | |
42 | negeq 7301 | . . . . . . . . . . 11 | |
43 | 42 | eqeq2d 2092 | . . . . . . . . . 10 |
44 | 43 | adantl 271 | . . . . . . . . 9 |
45 | recn 7106 | . . . . . . . . . . 11 | |
46 | negneg 7358 | . . . . . . . . . . . 12 | |
47 | 46 | eqcomd 2086 | . . . . . . . . . . 11 |
48 | 45, 47 | syl 14 | . . . . . . . . . 10 |
49 | 48 | ad2antrl 473 | . . . . . . . . 9 |
50 | 41, 44, 49 | rspcedvd 2708 | . . . . . . . 8 |
51 | 50 | ex 113 | . . . . . . 7 |
52 | 40, 51 | impbid 127 | . . . . . 6 |
53 | 52 | abbidv 2196 | . . . . 5 |
54 | 18 | rnmpt 4600 | . . . . 5 |
55 | df-rab 2357 | . . . . 5 | |
56 | 53, 54, 55 | 3eqtr4g 2138 | . . . 4 |
57 | 56 | eleq1d 2147 | . . 3 |
58 | 13, 23, 57 | 3bitrd 212 | . 2 |
59 | 58 | biimpa 290 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 cab 2067 wral 2348 wrex 2349 crab 2352 cvv 2601 wss 2973 cmpt 3839 cdm 4363 crn 4364 wfun 4916 wf1 4919 wf1o 4921 cfn 6244 cc 6979 cr 6980 cneg 7280 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-1o 6024 df-er 6129 df-en 6245 df-fin 6247 df-sub 7281 df-neg 7282 |
This theorem is referenced by: (None) |
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