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Mirrors > Home > MPE Home > Th. List > pnrmopn | Structured version Visualization version Unicode version |
Description: An open set in a perfectly normal space is a countable union of closed sets. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
pnrmopn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnrmtop 20406 |
. . . 4
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2 | eqid 2462 |
. . . . 5
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3 | 2 | opncld 20097 |
. . . 4
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4 | 1, 3 | sylan 478 |
. . 3
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5 | pnrmcld 20407 |
. . 3
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6 | 4, 5 | syldan 477 |
. 2
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7 | 1 | ad2antrr 737 |
. . . . . . . 8
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8 | elmapi 7519 |
. . . . . . . . . 10
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9 | 8 | adantl 472 |
. . . . . . . . 9
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10 | 9 | ffvelrnda 6045 |
. . . . . . . 8
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11 | 2 | opncld 20097 |
. . . . . . . 8
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12 | 7, 10, 11 | syl2anc 671 |
. . . . . . 7
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13 | eqid 2462 |
. . . . . . 7
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14 | 12, 13 | fmptd 6069 |
. . . . . 6
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15 | fvex 5898 |
. . . . . . 7
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16 | nnex 10643 |
. . . . . . 7
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17 | 15, 16 | elmap 7526 |
. . . . . 6
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18 | 14, 17 | sylibr 217 |
. . . . 5
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19 | iundif2 4359 |
. . . . . . 7
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20 | ffn 5751 |
. . . . . . . . 9
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21 | fniinfv 5947 |
. . . . . . . . 9
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22 | 9, 20, 21 | 3syl 18 |
. . . . . . . 8
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23 | 22 | difeq2d 3563 |
. . . . . . 7
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24 | 19, 23 | syl5eq 2508 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | uniexg 6615 |
. . . . . . . . . . 11
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26 | difexg 4565 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 25, 26 | syl 17 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 27 | ralrimivw 2815 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 28 | adantr 471 |
. . . . . . . 8
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30 | dfiun2g 4324 |
. . . . . . . 8
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31 | 29, 30 | syl 17 |
. . . . . . 7
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32 | 13 | rnmpt 5099 |
. . . . . . . 8
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33 | 32 | unieqi 4221 |
. . . . . . 7
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34 | 31, 33 | syl6eqr 2514 |
. . . . . 6
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35 | 24, 34 | eqtr3d 2498 |
. . . . 5
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36 | rneq 5079 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
37 | 36 | unieqd 4222 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | 37 | eqeq2d 2472 |
. . . . . 6
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39 | 38 | rspcev 3162 |
. . . . 5
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40 | 18, 35, 39 | syl2anc 671 |
. . . 4
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41 | 40 | ad2ant2r 758 |
. . 3
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42 | difeq2 3557 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
43 | 42 | eqcomd 2468 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
44 | elssuni 4241 |
. . . . . . . 8
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45 | dfss4 3689 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
46 | 44, 45 | sylib 201 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
47 | 43, 46 | sylan9eqr 2518 |
. . . . . 6
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48 | 47 | ad2ant2l 757 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 48 | eqeq1d 2464 |
. . . 4
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50 | 49 | rexbidv 2913 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
51 | 41, 50 | mpbid 215 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
52 | 6, 51 | rexlimddv 2895 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 ax-un 6610 ax-cnex 9621 ax-resscn 9622 ax-1cn 9623 ax-icn 9624 ax-addcl 9625 ax-addrcl 9626 ax-mulcl 9627 ax-mulrcl 9628 ax-i2m1 9633 ax-1ne0 9634 ax-rrecex 9637 ax-cnre 9638 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-reu 2756 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-pss 3432 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4213 df-int 4249 df-iun 4294 df-iin 4295 df-br 4417 df-opab 4476 df-mpt 4477 df-tr 4512 df-eprel 4764 df-id 4768 df-po 4774 df-so 4775 df-fr 4812 df-we 4814 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-rn 4864 df-res 4865 df-ima 4866 df-pred 5399 df-ord 5445 df-on 5446 df-lim 5447 df-suc 5448 df-iota 5565 df-fun 5603 df-fn 5604 df-f 5605 df-f1 5606 df-fo 5607 df-f1o 5608 df-fv 5609 df-ov 6318 df-oprab 6319 df-mpt2 6320 df-om 6720 df-1st 6820 df-2nd 6821 df-wrecs 7054 df-recs 7116 df-rdg 7154 df-map 7500 df-nn 10638 df-top 19970 df-cld 20083 df-nrm 20382 df-pnrm 20384 |
This theorem is referenced by: (None) |
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