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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 6708 | . . 3 | |
2 | 1 | biimpi 119 | . 2 |
3 | simplrr 526 | . . . . . . 7 | |
4 | simpr 109 | . . . . . . 7 | |
5 | 3, 4 | breqtrd 3992 | . . . . . 6 |
6 | en0 6742 | . . . . . 6 | |
7 | 5, 6 | sylib 121 | . . . . 5 |
8 | nner 2331 | . . . . 5 | |
9 | 7, 8 | syl 14 | . . . 4 |
10 | n0r 3408 | . . . . . 6 | |
11 | 10 | necon2bi 2382 | . . . . 5 |
12 | 7, 11 | syl 14 | . . . 4 |
13 | 9, 12 | 2falsed 692 | . . 3 |
14 | simplrr 526 | . . . . . . . . . 10 | |
15 | 14 | adantr 274 | . . . . . . . . 9 |
16 | 15 | ensymd 6730 | . . . . . . . 8 |
17 | bren 6694 | . . . . . . . 8 | |
18 | 16, 17 | sylib 121 | . . . . . . 7 |
19 | f1of 5416 | . . . . . . . . . . . 12 | |
20 | 19 | adantl 275 | . . . . . . . . . . 11 |
21 | sucidg 4378 | . . . . . . . . . . . . 13 | |
22 | 21 | ad3antlr 485 | . . . . . . . . . . . 12 |
23 | simplr 520 | . . . . . . . . . . . 12 | |
24 | 22, 23 | eleqtrrd 2237 | . . . . . . . . . . 11 |
25 | 20, 24 | ffvelrnd 5605 | . . . . . . . . . 10 |
26 | elex2 2728 | . . . . . . . . . 10 | |
27 | 25, 26 | syl 14 | . . . . . . . . 9 |
28 | 27, 10 | syl 14 | . . . . . . . 8 |
29 | 28, 27 | 2thd 174 | . . . . . . 7 |
30 | 18, 29 | exlimddv 1878 | . . . . . 6 |
31 | 30 | ex 114 | . . . . 5 |
32 | 31 | rexlimdva 2574 | . . . 4 |
33 | 32 | imp 123 | . . 3 |
34 | nn0suc 4565 | . . . 4 | |
35 | 34 | ad2antrl 482 | . . 3 |
36 | 13, 33, 35 | mpjaodan 788 | . 2 |
37 | 2, 36 | rexlimddv 2579 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 wceq 1335 wex 1472 wcel 2128 wne 2327 wrex 2436 c0 3395 class class class wbr 3967 csuc 4327 com 4551 wf 5168 wf1o 5171 cfv 5172 cen 6685 cfn 6687 |
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 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-iinf 4549 |
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-ne 2328 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 4028 df-id 4255 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-er 6482 df-en 6688 df-fin 6690 |
This theorem is referenced by: findcard2 6836 findcard2s 6837 diffisn 6840 fimax2gtri 6848 elfi2 6918 elfir 6919 fiuni 6924 fifo 6926 |
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