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Mirrors > Home > ILE Home > Th. List > unsnfi | Unicode version |
Description: Adding a singleton to a finite set yields a finite set. (Contributed by Jim Kingdon, 3-Feb-2022.) |
Ref | Expression |
---|---|
unsnfi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6330 |
. . . 4
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2 | 1 | biimpi 118 |
. . 3
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3 | 2 | 3ad2ant1 960 |
. 2
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4 | peano2 4365 |
. . . . 5
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5 | 4 | ad2antrl 474 |
. . . 4
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6 | simprr 499 |
. . . . . 6
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7 | simpl2 943 |
. . . . . . 7
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8 | simprl 498 |
. . . . . . 7
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9 | en2sn 6380 |
. . . . . . 7
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10 | 7, 8, 9 | syl2anc 403 |
. . . . . 6
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11 | disjsn 3473 |
. . . . . . . . 9
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12 | 11 | biimpri 131 |
. . . . . . . 8
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13 | 12 | 3ad2ant3 962 |
. . . . . . 7
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14 | 13 | adantr 270 |
. . . . . 6
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15 | nnord 4381 |
. . . . . . . . 9
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16 | ordirr 4314 |
. . . . . . . . 9
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17 | 15, 16 | syl 14 |
. . . . . . . 8
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18 | disjsn 3473 |
. . . . . . . 8
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19 | 17, 18 | sylibr 132 |
. . . . . . 7
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20 | 19 | ad2antrl 474 |
. . . . . 6
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21 | unen 6383 |
. . . . . 6
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22 | 6, 10, 14, 20, 21 | syl22anc 1171 |
. . . . 5
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23 | df-suc 4155 |
. . . . 5
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24 | 22, 23 | syl6breqr 3846 |
. . . 4
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25 | breq2 3810 |
. . . . 5
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26 | 25 | rspcev 2710 |
. . . 4
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27 | 5, 24, 26 | syl2anc 403 |
. . 3
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28 | isfi 6330 |
. . 3
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29 | 27, 28 | sylibr 132 |
. 2
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30 | 3, 29 | rexlimddv 2486 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3917 ax-nul 3925 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-iinf 4358 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-v 2612 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-nul 3269 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-int 3658 df-br 3807 df-opab 3861 df-tr 3897 df-id 4077 df-iord 4150 df-on 4152 df-suc 4155 df-iom 4361 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-rn 4403 df-res 4404 df-ima 4405 df-fun 4955 df-fn 4956 df-f 4957 df-f1 4958 df-fo 4959 df-f1o 4960 df-1o 6086 df-er 6194 df-en 6310 df-fin 6312 |
This theorem is referenced by: unfidisj 6467 fisseneq 6475 ssfirab 6476 fnfi 6479 |
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