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Mirrors > Home > MPE Home > Th. List > mreiincl | Structured version Visualization version Unicode version |
Description: A nonempty indexed intersection of closed sets is closed. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
mreiincl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiin2g 4311 |
. . 3
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2 | 1 | 3ad2ant3 1031 |
. 2
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3 | simp1 1008 |
. . 3
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4 | uniiunlem 3517 |
. . . . 5
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5 | 4 | ibi 245 |
. . . 4
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6 | 5 | 3ad2ant3 1031 |
. . 3
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7 | n0 3741 |
. . . . . 6
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8 | nfra1 2769 |
. . . . . . . 8
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9 | nfre1 2848 |
. . . . . . . . . 10
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10 | 9 | nfab 2596 |
. . . . . . . . 9
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11 | nfcv 2592 |
. . . . . . . . 9
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12 | 10, 11 | nfne 2723 |
. . . . . . . 8
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13 | 8, 12 | nfim 2003 |
. . . . . . 7
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14 | rsp 2754 |
. . . . . . . . . 10
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15 | 14 | com12 32 |
. . . . . . . . 9
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16 | elisset 3057 |
. . . . . . . . . . 11
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17 | rspe 2845 |
. . . . . . . . . . . 12
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18 | 17 | ex 436 |
. . . . . . . . . . 11
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19 | 16, 18 | syl5 33 |
. . . . . . . . . 10
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20 | rexcom4 3067 |
. . . . . . . . . 10
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21 | 19, 20 | syl6ib 230 |
. . . . . . . . 9
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22 | 15, 21 | syld 45 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | abn0 3751 |
. . . . . . . 8
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24 | 22, 23 | syl6ibr 231 |
. . . . . . 7
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25 | 13, 24 | exlimi 1995 |
. . . . . 6
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26 | 7, 25 | sylbi 199 |
. . . . 5
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27 | 26 | imp 431 |
. . . 4
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28 | 27 | 3adant1 1026 |
. . 3
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29 | mreintcl 15501 |
. . 3
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30 | 3, 6, 28, 29 | syl3anc 1268 |
. 2
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31 | 2, 30 | eqeltrd 2529 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-rab 2746 df-v 3047 df-sbc 3268 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-int 4235 df-iin 4281 df-br 4403 df-opab 4462 df-mpt 4463 df-id 4749 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-iota 5546 df-fun 5584 df-fv 5590 df-mre 15492 |
This theorem is referenced by: mreriincl 15504 mretopd 20108 |
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