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Mirrors > Home > MPE Home > Th. List > unfi | Unicode version |
Description: The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 16-Nov-2002.) |
Ref | Expression |
---|---|
unfi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diffi 7302 |
. 2
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2 | reeanv 2839 |
. . . 4
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3 | isfi 7094 |
. . . . 5
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4 | isfi 7094 |
. . . . 5
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5 | 3, 4 | anbi12i 679 |
. . . 4
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6 | 2, 5 | bitr4i 244 |
. . 3
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7 | nnacl 6817 |
. . . . 5
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8 | unfilem3 7336 |
. . . . . . 7
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9 | entr 7122 |
. . . . . . . 8
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10 | 9 | expcom 425 |
. . . . . . 7
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11 | 8, 10 | syl 16 |
. . . . . 6
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12 | disjdif 3664 |
. . . . . . . 8
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13 | disjdif 3664 |
. . . . . . . 8
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14 | unen 7152 |
. . . . . . . 8
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15 | 12, 13, 14 | mpanr12 667 |
. . . . . . 7
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16 | undif2 3668 |
. . . . . . . . 9
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17 | 16 | a1i 11 |
. . . . . . . 8
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18 | nnaword1 6835 |
. . . . . . . . 9
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19 | undif 3672 |
. . . . . . . . 9
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20 | 18, 19 | sylib 189 |
. . . . . . . 8
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21 | 17, 20 | breq12d 4189 |
. . . . . . 7
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22 | 15, 21 | syl5ib 211 |
. . . . . 6
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23 | 11, 22 | sylan2d 469 |
. . . . 5
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24 | breq2 4180 |
. . . . . . 7
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25 | 24 | rspcev 3016 |
. . . . . 6
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26 | isfi 7094 |
. . . . . 6
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27 | 25, 26 | sylibr 204 |
. . . . 5
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28 | 7, 23, 27 | ee12an 1369 |
. . . 4
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29 | 28 | rexlimivv 2799 |
. . 3
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30 | 6, 29 | sylbir 205 |
. 2
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31 | 1, 30 | sylan2 461 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: unfi2 7339 difinf 7340 xpfi 7341 prfi 7344 tpfi 7345 fnfi 7347 iunfi 7357 pwfilem 7363 fiin 7389 wemapso2 7481 cantnfp1lem1 7594 ficardun2 8043 ackbij1lem6 8065 ackbij1lem16 8075 fin23lem28 8180 fin23lem30 8182 isfin1-3 8226 axcclem 8297 hashun 11615 hashunlei 11643 hashmap 11657 hashbclem 11660 hashf1lem1 11663 hashf1lem2 11664 hashf1 11665 incexclem 12575 isumltss 12587 ramub1lem1 13353 fpwipodrs 14549 acsfiindd 14562 gsumzaddlem 15485 gsumunsn 15503 dprdfadd 15537 psrbagaddcl 16394 mplsubg 16459 mpllss 16460 fctop 17027 uncmp 17424 1stckgenlem 17542 ptbasin 17566 cfinfil 17882 fin1aufil 17921 alexsubALTlem3 18037 tmdgsum 18082 tsmsfbas 18114 tsmsgsum 18125 tsmsres 18130 tsmsxplem1 18139 prdsmet 18357 prdsbl 18478 icccmplem2 18811 ovolfiniun 19354 volfiniun 19398 fta1glem2 20046 fta1lem 20181 aannenlem2 20203 aalioulem2 20207 dchrfi 20996 usgrafilem2 21383 vdgrfiun 21630 konigsberg 21666 ballotlemgun 24739 itg2addnclem2 26160 locfincmp 26278 comppfsc 26281 prdsbnd 26396 funsnfsup 26637 elrfi 26642 mzpcompact2lem 26702 eldioph2 26714 lsmfgcl 27044 dsmmacl 27079 symgfisg 27281 fiuneneq 27385 pclfinN 30386 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2389 ax-sep 4294 ax-nul 4302 ax-pow 4341 ax-pr 4367 ax-un 4664 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2262 df-mo 2263 df-clab 2395 df-cleq 2401 df-clel 2404 df-nfc 2533 df-ne 2573 df-ral 2675 df-rex 2676 df-reu 2677 df-rab 2679 df-v 2922 df-sbc 3126 df-csb 3216 df-dif 3287 df-un 3289 df-in 3291 df-ss 3298 df-pss 3300 df-nul 3593 df-if 3704 df-pw 3765 df-sn 3784 df-pr 3785 df-tp 3786 df-op 3787 df-uni 3980 df-int 4015 df-iun 4059 df-br 4177 df-opab 4231 df-mpt 4232 df-tr 4267 df-eprel 4458 df-id 4462 df-po 4467 df-so 4468 df-fr 4505 df-we 4507 df-ord 4548 df-on 4549 df-lim 4550 df-suc 4551 df-om 4809 df-xp 4847 df-rel 4848 df-cnv 4849 df-co 4850 df-dm 4851 df-rn 4852 df-res 4853 df-ima 4854 df-iota 5381 df-fun 5419 df-fn 5420 df-f 5421 df-f1 5422 df-fo 5423 df-f1o 5424 df-fv 5425 df-ov 6047 df-oprab 6048 df-mpt2 6049 df-recs 6596 df-rdg 6631 df-oadd 6691 df-er 6868 df-en 7073 df-fin 7076 |
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