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Mirrors > Home > MPE Home > Th. List > mrieqvd | Structured version Visualization version GIF version |
Description: In a Moore system, a set is independent if and only if, for all elements of the set, the closure of the set with the element removed is unequal to the closure of the original set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mrieqvd.1 | β’ (π β π΄ β (Mooreβπ)) |
mrieqvd.2 | β’ π = (mrClsβπ΄) |
mrieqvd.3 | β’ πΌ = (mrIndβπ΄) |
mrieqvd.4 | β’ (π β π β π) |
Ref | Expression |
---|---|
mrieqvd | β’ (π β (π β πΌ β βπ₯ β π (πβ(π β {π₯})) β (πβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrieqvd.2 | . . 3 β’ π = (mrClsβπ΄) | |
2 | mrieqvd.3 | . . 3 β’ πΌ = (mrIndβπ΄) | |
3 | mrieqvd.1 | . . 3 β’ (π β π΄ β (Mooreβπ)) | |
4 | mrieqvd.4 | . . 3 β’ (π β π β π) | |
5 | 1, 2, 3, 4 | ismri2d 17576 | . 2 β’ (π β (π β πΌ β βπ₯ β π Β¬ π₯ β (πβ(π β {π₯})))) |
6 | 3 | adantr 481 | . . . . 5 β’ ((π β§ π₯ β π) β π΄ β (Mooreβπ)) |
7 | 4 | adantr 481 | . . . . 5 β’ ((π β§ π₯ β π) β π β π) |
8 | simpr 485 | . . . . 5 β’ ((π β§ π₯ β π) β π₯ β π) | |
9 | 6, 1, 7, 8 | mrieqvlemd 17572 | . . . 4 β’ ((π β§ π₯ β π) β (π₯ β (πβ(π β {π₯})) β (πβ(π β {π₯})) = (πβπ))) |
10 | 9 | necon3bbid 2978 | . . 3 β’ ((π β§ π₯ β π) β (Β¬ π₯ β (πβ(π β {π₯})) β (πβ(π β {π₯})) β (πβπ))) |
11 | 10 | ralbidva 3175 | . 2 β’ (π β (βπ₯ β π Β¬ π₯ β (πβ(π β {π₯})) β βπ₯ β π (πβ(π β {π₯})) β (πβπ))) |
12 | 5, 11 | bitrd 278 | 1 β’ (π β (π β πΌ β βπ₯ β π (πβ(π β {π₯})) β (πβπ))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2940 βwral 3061 β cdif 3945 β wss 3948 {csn 4628 βcfv 6543 Moorecmre 17525 mrClscmrc 17526 mrIndcmri 17527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-mre 17529 df-mrc 17530 df-mri 17531 |
This theorem is referenced by: (None) |
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