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Theorem mremre 15588
 Description: The Moore collections of subsets of a space, viewed as a kind of subset of the power set, form a Moore collection in their own right on the power set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
mremre Moore Moore

Proof of Theorem mremre
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mresspw 15576 . . . . 5 Moore
2 selpw 3949 . . . . 5
31, 2sylibr 217 . . . 4 Moore
43ssriv 3422 . . 3 Moore
54a1i 11 . 2 Moore
6 ssid 3437 . . . 4
76a1i 11 . . 3
8 pwidg 3955 . . 3
9 intssuni2 4251 . . . . . 6
1093adant1 1048 . . . . 5
11 unipw 4650 . . . . 5
1210, 11syl6sseq 3464 . . . 4
13 elpw2g 4564 . . . . 5
14133ad2ant1 1051 . . . 4
1512, 14mpbird 240 . . 3
167, 8, 15ismred 15586 . 2 Moore
17 n0 3732 . . . . 5
18 intss1 4241 . . . . . . . . 9
1918adantl 473 . . . . . . . 8 Moore
20 simpr 468 . . . . . . . . . 10 Moore Moore
2120sselda 3418 . . . . . . . . 9 Moore Moore
22 mresspw 15576 . . . . . . . . 9 Moore
2321, 22syl 17 . . . . . . . 8 Moore
2419, 23sstrd 3428 . . . . . . 7 Moore
2524ex 441 . . . . . 6 Moore
2625exlimdv 1787 . . . . 5 Moore
2717, 26syl5bi 225 . . . 4 Moore
28273impia 1228 . . 3 Moore
29 simp2 1031 . . . . . . 7 Moore Moore
3029sselda 3418 . . . . . 6 Moore Moore
31 mre1cl 15578 . . . . . 6 Moore
3230, 31syl 17 . . . . 5 Moore
3332ralrimiva 2809 . . . 4 Moore
34 elintg 4234 . . . . 5
35343ad2ant1 1051 . . . 4 Moore
3633, 35mpbird 240 . . 3 Moore
37 simp12 1061 . . . . . . 7 Moore Moore
3837sselda 3418 . . . . . 6 Moore Moore
39 simpl2 1034 . . . . . . 7 Moore
40 intss1 4241 . . . . . . . 8
4140adantl 473 . . . . . . 7 Moore
4239, 41sstrd 3428 . . . . . 6 Moore
43 simpl3 1035 . . . . . 6 Moore
44 mreintcl 15579 . . . . . 6 Moore
4538, 42, 43, 44syl3anc 1292 . . . . 5 Moore
4645ralrimiva 2809 . . . 4 Moore
47 intex 4557 . . . . . 6
48 elintg 4234 . . . . . 6
4947, 48sylbi 200 . . . . 5
50493ad2ant3 1053 . . . 4 Moore
5146, 50mpbird 240 . . 3 Moore
5228, 36, 51ismred 15586 . 2 Moore Moore
535, 16, 52ismred 15586 1 Moore Moore
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   w3a 1007  wex 1671   wcel 1904   wne 2641  wral 2756  cvv 3031   wss 3390  c0 3722  cpw 3942  cuni 4190  cint 4226  cfv 5589  Moorecmre 15566 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-mre 15570 This theorem is referenced by:  mreacs  15642  mreclatdemoBAD  20189
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