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Mirrors > Home > ILE Home > Th. List > fin0or | Unicode version |
Description: A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.) |
Ref | Expression |
---|---|
fin0or |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6241 | . . 3 | |
2 | 1 | biimpi 113 | . 2 |
3 | nn0suc 4327 | . . . 4 | |
4 | 3 | ad2antrl 459 | . . 3 |
5 | simplrr 488 | . . . . . . 7 | |
6 | simpr 103 | . . . . . . 7 | |
7 | 5, 6 | breqtrd 3788 | . . . . . 6 |
8 | en0 6275 | . . . . . 6 | |
9 | 7, 8 | sylib 127 | . . . . 5 |
10 | 9 | ex 108 | . . . 4 |
11 | simplrr 488 | . . . . . . . . . 10 | |
12 | 11 | adantr 261 | . . . . . . . . 9 |
13 | 12 | ensymd 6263 | . . . . . . . 8 |
14 | bren 6228 | . . . . . . . 8 | |
15 | 13, 14 | sylib 127 | . . . . . . 7 |
16 | f1of 5126 | . . . . . . . . . 10 | |
17 | 16 | adantl 262 | . . . . . . . . 9 |
18 | sucidg 4153 | . . . . . . . . . . 11 | |
19 | 18 | ad3antlr 462 | . . . . . . . . . 10 |
20 | simplr 482 | . . . . . . . . . 10 | |
21 | 19, 20 | eleqtrrd 2117 | . . . . . . . . 9 |
22 | 17, 21 | ffvelrnd 5303 | . . . . . . . 8 |
23 | elex2 2570 | . . . . . . . 8 | |
24 | 22, 23 | syl 14 | . . . . . . 7 |
25 | 15, 24 | exlimddv 1778 | . . . . . 6 |
26 | 25 | ex 108 | . . . . 5 |
27 | 26 | rexlimdva 2433 | . . . 4 |
28 | 10, 27 | orim12d 700 | . . 3 |
29 | 4, 28 | mpd 13 | . 2 |
30 | 2, 29 | rexlimddv 2437 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wo 629 wceq 1243 wex 1381 wcel 1393 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-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: (None) |
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