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Mirrors > Home > ILE Home > Th. List > iuncld | Unicode version |
Description: A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) (Revised by Jim Kingdon, 10-Mar-2023.) |
Ref | Expression |
---|---|
iuncld.1 |
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Ref | Expression |
---|---|
iuncld |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq1 3834 |
. . 3
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2 | 1 | eleq1d 2209 |
. 2
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3 | iuneq1 3834 |
. . 3
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4 | 3 | eleq1d 2209 |
. 2
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5 | iuneq1 3834 |
. . 3
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6 | 5 | eleq1d 2209 |
. 2
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7 | iuneq1 3834 |
. . 3
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8 | 7 | eleq1d 2209 |
. 2
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9 | 0iun 3878 |
. . . 4
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10 | 0cld 12320 |
. . . 4
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11 | 9, 10 | eqeltrid 2227 |
. . 3
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12 | 11 | 3ad2ant1 1003 |
. 2
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13 | simpr 109 |
. . . 4
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14 | nfcsb1v 3040 |
. . . . . . . 8
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15 | csbeq1a 3016 |
. . . . . . . 8
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16 | 14, 15 | iunxsngf 3898 |
. . . . . . 7
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17 | 16 | elv 2693 |
. . . . . 6
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18 | simprr 522 |
. . . . . . . 8
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19 | 18 | eldifad 3087 |
. . . . . . 7
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20 | simpll3 1023 |
. . . . . . 7
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21 | 14 | nfel1 2293 |
. . . . . . . 8
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22 | 15 | eleq1d 2209 |
. . . . . . . 8
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23 | 21, 22 | rspc 2787 |
. . . . . . 7
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24 | 19, 20, 23 | sylc 62 |
. . . . . 6
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25 | 17, 24 | eqeltrid 2227 |
. . . . 5
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26 | 25 | adantr 274 |
. . . 4
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27 | iunxun 3900 |
. . . . 5
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28 | uncld 12321 |
. . . . 5
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29 | 27, 28 | eqeltrid 2227 |
. . . 4
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30 | 13, 26, 29 | syl2anc 409 |
. . 3
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31 | 30 | ex 114 |
. 2
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32 | simp2 983 |
. 2
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33 | 2, 4, 6, 8, 12, 31, 32 | findcard2sd 6794 |
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-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 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-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 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 df-top 12204 df-cld 12303 |
This theorem is referenced by: unicld 12324 |
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