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Mirrors > Home > ILE Home > Th. List > fin0 | Unicode version |
Description: A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.) |
Ref | Expression |
---|---|
fin0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6241 | . . 3 | |
2 | 1 | biimpi 113 | . 2 |
3 | simplrr 488 | . . . . . . 7 | |
4 | simpr 103 | . . . . . . 7 | |
5 | 3, 4 | breqtrd 3788 | . . . . . 6 |
6 | en0 6275 | . . . . . 6 | |
7 | 5, 6 | sylib 127 | . . . . 5 |
8 | nner 2210 | . . . . 5 | |
9 | 7, 8 | syl 14 | . . . 4 |
10 | n0r 3234 | . . . . . 6 | |
11 | 10 | necon2bi 2260 | . . . . 5 |
12 | 7, 11 | syl 14 | . . . 4 |
13 | 9, 12 | 2falsed 618 | . . 3 |
14 | simplrr 488 | . . . . . . . . . 10 | |
15 | 14 | adantr 261 | . . . . . . . . 9 |
16 | 15 | ensymd 6263 | . . . . . . . 8 |
17 | bren 6228 | . . . . . . . 8 | |
18 | 16, 17 | sylib 127 | . . . . . . 7 |
19 | f1of 5126 | . . . . . . . . . . . 12 | |
20 | 19 | adantl 262 | . . . . . . . . . . 11 |
21 | sucidg 4153 | . . . . . . . . . . . . 13 | |
22 | 21 | ad3antlr 462 | . . . . . . . . . . . 12 |
23 | simplr 482 | . . . . . . . . . . . 12 | |
24 | 22, 23 | eleqtrrd 2117 | . . . . . . . . . . 11 |
25 | 20, 24 | ffvelrnd 5303 | . . . . . . . . . 10 |
26 | elex2 2570 | . . . . . . . . . 10 | |
27 | 25, 26 | syl 14 | . . . . . . . . 9 |
28 | 27, 10 | syl 14 | . . . . . . . 8 |
29 | 28, 27 | 2thd 164 | . . . . . . 7 |
30 | 18, 29 | exlimddv 1778 | . . . . . 6 |
31 | 30 | ex 108 | . . . . 5 |
32 | 31 | rexlimdva 2433 | . . . 4 |
33 | 32 | imp 115 | . . 3 |
34 | nn0suc 4327 | . . . 4 | |
35 | 34 | ad2antrl 459 | . . 3 |
36 | 13, 33, 35 | mpjaodan 711 | . 2 |
37 | 2, 36 | rexlimddv 2437 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 wceq 1243 wex 1381 wcel 1393 wne 2204 wrex 2307 c0 3224 class class class wbr 3764 csuc 4102 com 4313 wf 4898 wf1o 4901 cfv 4902 cen 6219 cfn 6221 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-id 4030 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-er 6106 df-en 6222 df-fin 6224 |
This theorem is referenced by: findcard2 6346 findcard2s 6347 diffisn 6350 |
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