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Mirrors > Home > MPE Home > Th. List > mreintcl | Structured version Visualization version GIF version |
Description: A nonempty collection of closed sets has a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
mreintcl | β’ ((πΆ β (Mooreβπ) β§ π β πΆ β§ π β β ) β β© π β πΆ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpw2g 5334 | . . . 4 β’ (πΆ β (Mooreβπ) β (π β π« πΆ β π β πΆ)) | |
2 | 1 | biimpar 477 | . . 3 β’ ((πΆ β (Mooreβπ) β§ π β πΆ) β π β π« πΆ) |
3 | 2 | 3adant3 1129 | . 2 β’ ((πΆ β (Mooreβπ) β§ π β πΆ β§ π β β ) β π β π« πΆ) |
4 | ismre 17530 | . . . 4 β’ (πΆ β (Mooreβπ) β (πΆ β π« π β§ π β πΆ β§ βπ β π« πΆ(π β β β β© π β πΆ))) | |
5 | 4 | simp3bi 1144 | . . 3 β’ (πΆ β (Mooreβπ) β βπ β π« πΆ(π β β β β© π β πΆ)) |
6 | 5 | 3ad2ant1 1130 | . 2 β’ ((πΆ β (Mooreβπ) β§ π β πΆ β§ π β β ) β βπ β π« πΆ(π β β β β© π β πΆ)) |
7 | simp3 1135 | . 2 β’ ((πΆ β (Mooreβπ) β§ π β πΆ β§ π β β ) β π β β ) | |
8 | neeq1 2995 | . . . . 5 β’ (π = π β (π β β β π β β )) | |
9 | inteq 4943 | . . . . . 6 β’ (π = π β β© π = β© π) | |
10 | 9 | eleq1d 2810 | . . . . 5 β’ (π = π β (β© π β πΆ β β© π β πΆ)) |
11 | 8, 10 | imbi12d 344 | . . . 4 β’ (π = π β ((π β β β β© π β πΆ) β (π β β β β© π β πΆ))) |
12 | 11 | rspcva 3602 | . . 3 β’ ((π β π« πΆ β§ βπ β π« πΆ(π β β β β© π β πΆ)) β (π β β β β© π β πΆ)) |
13 | 12 | 3impia 1114 | . 2 β’ ((π β π« πΆ β§ βπ β π« πΆ(π β β β β© π β πΆ) β§ π β β ) β β© π β πΆ) |
14 | 3, 6, 7, 13 | syl3anc 1368 | 1 β’ ((πΆ β (Mooreβπ) β§ π β πΆ β§ π β β ) β β© π β πΆ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2932 βwral 3053 β wss 3940 β c0 4314 π« cpw 4594 β© cint 4940 βcfv 6533 Moorecmre 17522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6485 df-fun 6535 df-fv 6541 df-mre 17526 |
This theorem is referenced by: mreiincl 17536 mrerintcl 17537 mreincl 17539 mremre 17544 submre 17545 mrcflem 17546 mrelatglb 18512 mreclatBAD 18515 mrelatglbALT 47775 mreclat 47776 |
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