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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 3283 |
. . 3
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2 | 1 | eleq1d 2246 |
. 2
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3 | uneq2 3283 |
. . 3
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4 | 3 | eleq1d 2246 |
. 2
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5 | uneq2 3283 |
. . 3
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6 | 5 | eleq1d 2246 |
. 2
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7 | uneq2 3283 |
. . 3
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8 | 7 | eleq1d 2246 |
. 2
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9 | un0 3456 |
. . 3
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10 | simp1 997 |
. . 3
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11 | 9, 10 | eqeltrid 2264 |
. 2
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12 | unass 3292 |
. . . 4
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13 | simpr 110 |
. . . . 5
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14 | vex 2740 |
. . . . . 6
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15 | 14 | a1i 9 |
. . . . 5
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16 | simplrr 536 |
. . . . . . . . 9
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17 | 16 | eldifad 3140 |
. . . . . . . 8
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18 | simp3 999 |
. . . . . . . . 9
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19 | 18 | ad3antrrr 492 |
. . . . . . . 8
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20 | minel 3484 |
. . . . . . . 8
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21 | 17, 19, 20 | syl2anc 411 |
. . . . . . 7
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22 | 16 | eldifbd 3141 |
. . . . . . 7
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23 | ioran 752 |
. . . . . . 7
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24 | 21, 22, 23 | sylanbrc 417 |
. . . . . 6
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25 | elun 3276 |
. . . . . 6
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26 | 24, 25 | sylnibr 677 |
. . . . 5
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27 | unsnfi 6911 |
. . . . 5
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28 | 13, 15, 26, 27 | syl3anc 1238 |
. . . 4
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29 | 12, 28 | eqeltrrid 2265 |
. . 3
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30 | 29 | ex 115 |
. 2
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31 | simp2 998 |
. 2
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32 | 2, 4, 6, 8, 11, 30, 31 | findcard2sd 6885 |
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 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-iord 4362 df-on 4364 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-1o 6410 df-er 6528 df-en 6734 df-fin 6736 |
This theorem is referenced by: unfiin 6918 prfidisj 6919 tpfidisj 6920 xpfi 6922 iunfidisj 6938 hashunlem 10755 hashun 10756 fsumsplitsnun 11398 fsum2dlemstep 11413 fsumconst 11433 fprodsplitsn 11612 |
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