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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 6739 | . . 3 | |
2 | 1 | biimpi 119 | . 2 |
3 | nn0suc 4588 | . . . 4 | |
4 | 3 | ad2antrl 487 | . . 3 |
5 | simplrr 531 | . . . . . . 7 | |
6 | simpr 109 | . . . . . . 7 | |
7 | 5, 6 | breqtrd 4015 | . . . . . 6 |
8 | en0 6773 | . . . . . 6 | |
9 | 7, 8 | sylib 121 | . . . . 5 |
10 | 9 | ex 114 | . . . 4 |
11 | simplrr 531 | . . . . . . . . . 10 | |
12 | 11 | adantr 274 | . . . . . . . . 9 |
13 | 12 | ensymd 6761 | . . . . . . . 8 |
14 | bren 6725 | . . . . . . . 8 | |
15 | 13, 14 | sylib 121 | . . . . . . 7 |
16 | f1of 5442 | . . . . . . . . . 10 | |
17 | 16 | adantl 275 | . . . . . . . . 9 |
18 | sucidg 4401 | . . . . . . . . . . 11 | |
19 | 18 | ad3antlr 490 | . . . . . . . . . 10 |
20 | simplr 525 | . . . . . . . . . 10 | |
21 | 19, 20 | eleqtrrd 2250 | . . . . . . . . 9 |
22 | 17, 21 | ffvelrnd 5632 | . . . . . . . 8 |
23 | elex2 2746 | . . . . . . . 8 | |
24 | 22, 23 | syl 14 | . . . . . . 7 |
25 | 15, 24 | exlimddv 1891 | . . . . . 6 |
26 | 25 | ex 114 | . . . . 5 |
27 | 26 | rexlimdva 2587 | . . . 4 |
28 | 10, 27 | orim12d 781 | . . 3 |
29 | 4, 28 | mpd 13 | . 2 |
30 | 2, 29 | rexlimddv 2592 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 703 wceq 1348 wex 1485 wcel 2141 wrex 2449 c0 3414 class class class wbr 3989 csuc 4350 com 4574 wf 5194 wf1o 5197 cfv 5198 cen 6716 cfn 6718 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-er 6513 df-en 6719 df-fin 6721 |
This theorem is referenced by: xpfi 6907 fival 6947 fiubm 10763 fsumcllem 11362 fprodcllem 11569 |
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