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Mirrors > Home > MPE Home > Th. List > negfi | Structured version Visualization version 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 3597 | . . . . . . . . . 10 | |
2 | renegcl 10344 | . . . . . . . . . 10 | |
3 | 1, 2 | syl6 35 | . . . . . . . . 9 |
4 | 3 | imp 445 | . . . . . . . 8 |
5 | 4 | ralrimiva 2966 | . . . . . . 7 |
6 | dmmptg 5632 | . . . . . . 7 | |
7 | 5, 6 | syl 17 | . . . . . 6 |
8 | 7 | eqcomd 2628 | . . . . 5 |
9 | 8 | eleq1d 2686 | . . . 4 |
10 | funmpt 5926 | . . . . 5 | |
11 | fundmfibi 8245 | . . . . 5 | |
12 | 10, 11 | mp1i 13 | . . . 4 |
13 | 9, 12 | bitr4d 271 | . . 3 |
14 | reex 10027 | . . . . . 6 | |
15 | 14 | ssex 4802 | . . . . 5 |
16 | mptexg 6484 | . . . . 5 | |
17 | 15, 16 | syl 17 | . . . 4 |
18 | eqid 2622 | . . . . . 6 | |
19 | 18 | negf1o 10460 | . . . . 5 |
20 | f1of1 6136 | . . . . 5 | |
21 | 19, 20 | syl 17 | . . . 4 |
22 | f1vrnfibi 8251 | . . . 4 | |
23 | 17, 21, 22 | syl2anc 693 | . . 3 |
24 | 1 | imp 445 | . . . . . . . . . 10 |
25 | 2 | adantl 482 | . . . . . . . . . . 11 |
26 | recn 10026 | . . . . . . . . . . . . . . . . 17 | |
27 | 26 | negnegd 10383 | . . . . . . . . . . . . . . . 16 |
28 | 27 | eqcomd 2628 | . . . . . . . . . . . . . . 15 |
29 | 28 | eleq1d 2686 | . . . . . . . . . . . . . 14 |
30 | 29 | biimpcd 239 | . . . . . . . . . . . . 13 |
31 | 30 | adantl 482 | . . . . . . . . . . . 12 |
32 | 31 | imp 445 | . . . . . . . . . . 11 |
33 | 25, 32 | jca 554 | . . . . . . . . . 10 |
34 | 24, 33 | mpdan 702 | . . . . . . . . 9 |
35 | eleq1 2689 | . . . . . . . . . 10 | |
36 | negeq 10273 | . . . . . . . . . . 11 | |
37 | 36 | eleq1d 2686 | . . . . . . . . . 10 |
38 | 35, 37 | anbi12d 747 | . . . . . . . . 9 |
39 | 34, 38 | syl5ibrcom 237 | . . . . . . . 8 |
40 | 39 | rexlimdva 3031 | . . . . . . 7 |
41 | simprr 796 | . . . . . . . . 9 | |
42 | negeq 10273 | . . . . . . . . . . 11 | |
43 | 42 | eqeq2d 2632 | . . . . . . . . . 10 |
44 | 43 | adantl 482 | . . . . . . . . 9 |
45 | recn 10026 | . . . . . . . . . . 11 | |
46 | negneg 10331 | . . . . . . . . . . . 12 | |
47 | 46 | eqcomd 2628 | . . . . . . . . . . 11 |
48 | 45, 47 | syl 17 | . . . . . . . . . 10 |
49 | 48 | ad2antrl 764 | . . . . . . . . 9 |
50 | 41, 44, 49 | rspcedvd 3317 | . . . . . . . 8 |
51 | 50 | ex 450 | . . . . . . 7 |
52 | 40, 51 | impbid 202 | . . . . . 6 |
53 | 52 | abbidv 2741 | . . . . 5 |
54 | 18 | rnmpt 5371 | . . . . 5 |
55 | df-rab 2921 | . . . . 5 | |
56 | 53, 54, 55 | 3eqtr4g 2681 | . . . 4 |
57 | 56 | eleq1d 2686 | . . 3 |
58 | 13, 23, 57 | 3bitrd 294 | . 2 |
59 | 58 | biimpa 501 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 cab 2608 wral 2912 wrex 2913 crab 2916 cvv 3200 wss 3574 cmpt 4729 cdm 5114 crn 5115 wfun 5882 wf1 5885 wf1o 5887 cfn 7955 cc 9934 cr 9935 cneg 10267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 |
This theorem is referenced by: fiminre 10972 |
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