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Theorem mremre 13821
 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 13809 . . . . 5 Moore
2 vex 2951 . . . . . 6
32elpw 3797 . . . . 5
41, 3sylibr 204 . . . 4 Moore
54ssriv 3344 . . 3 Moore
65a1i 11 . 2 Moore
7 ssid 3359 . . . 4
87a1i 11 . . 3
9 pwidg 3803 . . 3
10 intssuni2 4067 . . . . . 6
11103adant1 975 . . . . 5
12 unipw 4406 . . . . 5
1311, 12syl6sseq 3386 . . . 4
14 elpw2g 4355 . . . . 5
15143ad2ant1 978 . . . 4
1613, 15mpbird 224 . . 3
178, 9, 16ismred 13819 . 2 Moore
18 n0 3629 . . . . 5
19 intss1 4057 . . . . . . . . 9
2019adantl 453 . . . . . . . 8 Moore
21 simpr 448 . . . . . . . . . 10 Moore Moore
2221sselda 3340 . . . . . . . . 9 Moore Moore
23 mresspw 13809 . . . . . . . . 9 Moore
2422, 23syl 16 . . . . . . . 8 Moore
2520, 24sstrd 3350 . . . . . . 7 Moore
2625ex 424 . . . . . 6 Moore
2726exlimdv 1646 . . . . 5 Moore
2818, 27syl5bi 209 . . . 4 Moore
29283impia 1150 . . 3 Moore
30 simp2 958 . . . . . . 7 Moore Moore
3130sselda 3340 . . . . . 6 Moore Moore
32 mre1cl 13811 . . . . . 6 Moore
3331, 32syl 16 . . . . 5 Moore
3433ralrimiva 2781 . . . 4 Moore
35 elintg 4050 . . . . 5
36353ad2ant1 978 . . . 4 Moore
3734, 36mpbird 224 . . 3 Moore
38 simp12 988 . . . . . . 7 Moore Moore
3938sselda 3340 . . . . . 6 Moore Moore
40 simpl2 961 . . . . . . 7 Moore
41 intss1 4057 . . . . . . . 8
4241adantl 453 . . . . . . 7 Moore
4340, 42sstrd 3350 . . . . . 6 Moore
44 simpl3 962 . . . . . 6 Moore
45 mreintcl 13812 . . . . . 6 Moore
4639, 43, 44, 45syl3anc 1184 . . . . 5 Moore
4746ralrimiva 2781 . . . 4 Moore
48 intex 4348 . . . . . 6
49 elintg 4050 . . . . . 6
5048, 49sylbi 188 . . . . 5
51503ad2ant3 980 . . . 4 Moore
5247, 51mpbird 224 . . 3 Moore
5329, 37, 52ismred 13819 . 2 Moore Moore
546, 17, 53ismred 13819 1 Moore Moore
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936  wex 1550   wcel 1725   wne 2598  wral 2697  cvv 2948   wss 3312  c0 3620  cpw 3791  cuni 4007  cint 4042  cfv 5446  Moorecmre 13799 This theorem is referenced by:  mreacs  13875  mreclatdemo  17152 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-mre 13803
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