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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 3131 | . . . . . . . . . 10 | |
2 | renegcl 8150 | . . . . . . . . . 10 | |
3 | 1, 2 | syl6 33 | . . . . . . . . 9 |
4 | 3 | imp 123 | . . . . . . . 8 |
5 | 4 | ralrimiva 2537 | . . . . . . 7 |
6 | dmmptg 5095 | . . . . . . 7 | |
7 | 5, 6 | syl 14 | . . . . . 6 |
8 | 7 | eqcomd 2170 | . . . . 5 |
9 | 8 | eleq1d 2233 | . . . 4 |
10 | funmpt 5220 | . . . . 5 | |
11 | fundmfibi 6895 | . . . . 5 | |
12 | 10, 11 | mp1i 10 | . . . 4 |
13 | 9, 12 | bitr4d 190 | . . 3 |
14 | reex 7878 | . . . . . 6 | |
15 | 14 | ssex 4113 | . . . . 5 |
16 | mptexg 5704 | . . . . 5 | |
17 | 15, 16 | syl 14 | . . . 4 |
18 | eqid 2164 | . . . . . 6 | |
19 | 18 | negf1o 8271 | . . . . 5 |
20 | f1of1 5425 | . . . . 5 | |
21 | 19, 20 | syl 14 | . . . 4 |
22 | f1vrnfibi 6901 | . . . 4 | |
23 | 17, 21, 22 | syl2anc 409 | . . 3 |
24 | 1 | imp 123 | . . . . . . . . . 10 |
25 | 2 | adantl 275 | . . . . . . . . . . 11 |
26 | recn 7877 | . . . . . . . . . . . . . . . . 17 | |
27 | 26 | negnegd 8191 | . . . . . . . . . . . . . . . 16 |
28 | 27 | eqcomd 2170 | . . . . . . . . . . . . . . 15 |
29 | 28 | eleq1d 2233 | . . . . . . . . . . . . . 14 |
30 | 29 | biimpcd 158 | . . . . . . . . . . . . 13 |
31 | 30 | adantl 275 | . . . . . . . . . . . 12 |
32 | 31 | imp 123 | . . . . . . . . . . 11 |
33 | 25, 32 | jca 304 | . . . . . . . . . 10 |
34 | 24, 33 | mpdan 418 | . . . . . . . . 9 |
35 | eleq1 2227 | . . . . . . . . . 10 | |
36 | negeq 8082 | . . . . . . . . . . 11 | |
37 | 36 | eleq1d 2233 | . . . . . . . . . 10 |
38 | 35, 37 | anbi12d 465 | . . . . . . . . 9 |
39 | 34, 38 | syl5ibrcom 156 | . . . . . . . 8 |
40 | 39 | rexlimdva 2581 | . . . . . . 7 |
41 | simprr 522 | . . . . . . . . 9 | |
42 | negeq 8082 | . . . . . . . . . . 11 | |
43 | 42 | eqeq2d 2176 | . . . . . . . . . 10 |
44 | 43 | adantl 275 | . . . . . . . . 9 |
45 | recn 7877 | . . . . . . . . . . 11 | |
46 | negneg 8139 | . . . . . . . . . . . 12 | |
47 | 46 | eqcomd 2170 | . . . . . . . . . . 11 |
48 | 45, 47 | syl 14 | . . . . . . . . . 10 |
49 | 48 | ad2antrl 482 | . . . . . . . . 9 |
50 | 41, 44, 49 | rspcedvd 2831 | . . . . . . . 8 |
51 | 50 | ex 114 | . . . . . . 7 |
52 | 40, 51 | impbid 128 | . . . . . 6 |
53 | 52 | abbidv 2282 | . . . . 5 |
54 | 18 | rnmpt 4846 | . . . . 5 |
55 | df-rab 2451 | . . . . 5 | |
56 | 53, 54, 55 | 3eqtr4g 2222 | . . . 4 |
57 | 56 | eleq1d 2233 | . . 3 |
58 | 13, 23, 57 | 3bitrd 213 | . 2 |
59 | 58 | biimpa 294 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 cab 2150 wral 2442 wrex 2443 crab 2446 cvv 2721 wss 3111 cmpt 4037 cdm 4598 crn 4599 wfun 5176 wf1 5179 wf1o 5181 cfn 6697 cc 7742 cr 7743 cneg 8061 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-1o 6375 df-er 6492 df-en 6698 df-fin 6700 df-sub 8062 df-neg 8063 |
This theorem is referenced by: (None) |
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