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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 6709 | . . 3 | |
2 | 1 | biimpi 119 | . 2 |
3 | nn0suc 4566 | . . . 4 | |
4 | 3 | ad2antrl 482 | . . 3 |
5 | simplrr 526 | . . . . . . 7 | |
6 | simpr 109 | . . . . . . 7 | |
7 | 5, 6 | breqtrd 3993 | . . . . . 6 |
8 | en0 6743 | . . . . . 6 | |
9 | 7, 8 | sylib 121 | . . . . 5 |
10 | 9 | ex 114 | . . . 4 |
11 | simplrr 526 | . . . . . . . . . 10 | |
12 | 11 | adantr 274 | . . . . . . . . 9 |
13 | 12 | ensymd 6731 | . . . . . . . 8 |
14 | bren 6695 | . . . . . . . 8 | |
15 | 13, 14 | sylib 121 | . . . . . . 7 |
16 | f1of 5417 | . . . . . . . . . 10 | |
17 | 16 | adantl 275 | . . . . . . . . 9 |
18 | sucidg 4379 | . . . . . . . . . . 11 | |
19 | 18 | ad3antlr 485 | . . . . . . . . . 10 |
20 | simplr 520 | . . . . . . . . . 10 | |
21 | 19, 20 | eleqtrrd 2237 | . . . . . . . . 9 |
22 | 17, 21 | ffvelrnd 5606 | . . . . . . . 8 |
23 | elex2 2728 | . . . . . . . 8 | |
24 | 22, 23 | syl 14 | . . . . . . 7 |
25 | 15, 24 | exlimddv 1878 | . . . . . 6 |
26 | 25 | ex 114 | . . . . 5 |
27 | 26 | rexlimdva 2574 | . . . 4 |
28 | 10, 27 | orim12d 776 | . . 3 |
29 | 4, 28 | mpd 13 | . 2 |
30 | 2, 29 | rexlimddv 2579 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 wceq 1335 wex 1472 wcel 2128 wrex 2436 c0 3395 class class class wbr 3967 csuc 4328 com 4552 wf 5169 wf1o 5172 cfv 5173 cen 6686 cfn 6688 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-nul 4093 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-iinf 4550 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4029 df-id 4256 df-suc 4334 df-iom 4553 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-er 6483 df-en 6689 df-fin 6691 |
This theorem is referenced by: xpfi 6877 fival 6917 fsumcllem 11308 fprodcllem 11515 |
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