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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6663 |
. . 3
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2 | 1 | biimpi 119 |
. 2
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3 | nn0suc 4526 |
. . . 4
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4 | 3 | ad2antrl 482 |
. . 3
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5 | simplrr 526 |
. . . . . . 7
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6 | simpr 109 |
. . . . . . 7
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7 | 5, 6 | breqtrd 3962 |
. . . . . 6
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8 | en0 6697 |
. . . . . 6
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9 | 7, 8 | sylib 121 |
. . . . 5
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10 | 9 | ex 114 |
. . . 4
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11 | simplrr 526 |
. . . . . . . . . 10
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12 | 11 | adantr 274 |
. . . . . . . . 9
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13 | 12 | ensymd 6685 |
. . . . . . . 8
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14 | bren 6649 |
. . . . . . . 8
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15 | 13, 14 | sylib 121 |
. . . . . . 7
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16 | f1of 5375 |
. . . . . . . . . 10
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17 | 16 | adantl 275 |
. . . . . . . . 9
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18 | sucidg 4346 |
. . . . . . . . . . 11
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19 | 18 | ad3antlr 485 |
. . . . . . . . . 10
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20 | simplr 520 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 19, 20 | eleqtrrd 2220 |
. . . . . . . . 9
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22 | 17, 21 | ffvelrnd 5564 |
. . . . . . . 8
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23 | elex2 2705 |
. . . . . . . 8
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24 | 22, 23 | syl 14 |
. . . . . . 7
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25 | 15, 24 | exlimddv 1871 |
. . . . . 6
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26 | 25 | ex 114 |
. . . . 5
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27 | 26 | rexlimdva 2552 |
. . . 4
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28 | 10, 27 | orim12d 776 |
. . 3
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29 | 4, 28 | mpd 13 |
. 2
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30 | 2, 29 | rexlimddv 2557 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-er 6437 df-en 6643 df-fin 6645 |
This theorem is referenced by: xpfi 6826 fival 6866 fsumcllem 11200 |
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