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Mirrors > Home > ILE Home > Th. List > unfidisj | Unicode version |
Description: The union of two disjoint finite sets is finite. (Contributed by Jim Kingdon, 25-Feb-2022.) |
Ref | Expression |
---|---|
unfidisj |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq2 3190 |
. . 3
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2 | 1 | eleq1d 2183 |
. 2
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3 | uneq2 3190 |
. . 3
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4 | 3 | eleq1d 2183 |
. 2
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5 | uneq2 3190 |
. . 3
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6 | 5 | eleq1d 2183 |
. 2
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7 | uneq2 3190 |
. . 3
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8 | 7 | eleq1d 2183 |
. 2
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9 | un0 3362 |
. . 3
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10 | simp1 964 |
. . 3
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11 | 9, 10 | syl5eqel 2201 |
. 2
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12 | unass 3199 |
. . . 4
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13 | simpr 109 |
. . . . 5
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14 | vex 2660 |
. . . . . 6
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15 | 14 | a1i 9 |
. . . . 5
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16 | simplrr 508 |
. . . . . . . . 9
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17 | 16 | eldifad 3048 |
. . . . . . . 8
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18 | simp3 966 |
. . . . . . . . 9
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19 | 18 | ad3antrrr 481 |
. . . . . . . 8
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20 | minel 3390 |
. . . . . . . 8
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21 | 17, 19, 20 | syl2anc 406 |
. . . . . . 7
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22 | 16 | eldifbd 3049 |
. . . . . . 7
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23 | ioran 724 |
. . . . . . 7
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24 | 21, 22, 23 | sylanbrc 411 |
. . . . . 6
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25 | elun 3183 |
. . . . . 6
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26 | 24, 25 | sylnibr 649 |
. . . . 5
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27 | unsnfi 6760 |
. . . . 5
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28 | 13, 15, 26, 27 | syl3anc 1199 |
. . . 4
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29 | 12, 28 | syl5eqelr 2202 |
. . 3
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30 | 29 | ex 114 |
. 2
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31 | simp2 965 |
. 2
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32 | 2, 4, 6, 8, 11, 30, 31 | findcard2sd 6739 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-coll 4003 ax-sep 4006 ax-nul 4014 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-iinf 4462 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-if 3441 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-iun 3781 df-br 3896 df-opab 3950 df-mpt 3951 df-tr 3987 df-id 4175 df-iord 4248 df-on 4250 df-suc 4253 df-iom 4465 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 df-fv 5089 df-1o 6267 df-er 6383 df-en 6589 df-fin 6591 |
This theorem is referenced by: unfiin 6767 prfidisj 6768 tpfidisj 6769 xpfi 6771 iunfidisj 6786 hashunlem 10440 hashun 10441 fsumsplitsnun 11077 fsum2dlemstep 11092 fsumconst 11112 |
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