Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6655 | . . 3 | |
2 | 1 | biimpi 119 | . 2 |
3 | nn0suc 4518 | . . . 4 | |
4 | 3 | ad2antrl 481 | . . 3 |
5 | simplrr 525 | . . . . . . 7 | |
6 | simpr 109 | . . . . . . 7 | |
7 | 5, 6 | breqtrd 3954 | . . . . . 6 |
8 | en0 6689 | . . . . . 6 | |
9 | 7, 8 | sylib 121 | . . . . 5 |
10 | 9 | ex 114 | . . . 4 |
11 | simplrr 525 | . . . . . . . . . 10 | |
12 | 11 | adantr 274 | . . . . . . . . 9 |
13 | 12 | ensymd 6677 | . . . . . . . 8 |
14 | bren 6641 | . . . . . . . 8 | |
15 | 13, 14 | sylib 121 | . . . . . . 7 |
16 | f1of 5367 | . . . . . . . . . 10 | |
17 | 16 | adantl 275 | . . . . . . . . 9 |
18 | sucidg 4338 | . . . . . . . . . . 11 | |
19 | 18 | ad3antlr 484 | . . . . . . . . . 10 |
20 | simplr 519 | . . . . . . . . . 10 | |
21 | 19, 20 | eleqtrrd 2219 | . . . . . . . . 9 |
22 | 17, 21 | ffvelrnd 5556 | . . . . . . . 8 |
23 | elex2 2702 | . . . . . . . 8 | |
24 | 22, 23 | syl 14 | . . . . . . 7 |
25 | 15, 24 | exlimddv 1870 | . . . . . 6 |
26 | 25 | ex 114 | . . . . 5 |
27 | 26 | rexlimdva 2549 | . . . 4 |
28 | 10, 27 | orim12d 775 | . . 3 |
29 | 4, 28 | mpd 13 | . 2 |
30 | 2, 29 | rexlimddv 2554 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 wceq 1331 wex 1468 wcel 1480 wrex 2417 c0 3363 class class class wbr 3929 csuc 4287 com 4504 wf 5119 wf1o 5122 cfv 5123 cen 6632 cfn 6634 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-er 6429 df-en 6635 df-fin 6637 |
This theorem is referenced by: xpfi 6818 fival 6858 fsumcllem 11168 |
Copyright terms: Public domain | W3C validator |