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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 17584 | . 2 β’ (π β (π β πΌ β βπ₯ β π Β¬ π₯ β (πβ(π β {π₯})))) |
6 | 3 | adantr 480 | . . . . 5 β’ ((π β§ π₯ β π) β π΄ β (Mooreβπ)) |
7 | 4 | adantr 480 | . . . . 5 β’ ((π β§ π₯ β π) β π β π) |
8 | simpr 484 | . . . . 5 β’ ((π β§ π₯ β π) β π₯ β π) | |
9 | 6, 1, 7, 8 | mrieqvlemd 17580 | . . . 4 β’ ((π β§ π₯ β π) β (π₯ β (πβ(π β {π₯})) β (πβ(π β {π₯})) = (πβπ))) |
10 | 9 | necon3bbid 2972 | . . 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 3169 | . 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 279 | 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 395 = wceq 1533 β wcel 2098 β wne 2934 βwral 3055 β cdif 3940 β wss 3943 {csn 4623 βcfv 6536 Moorecmre 17533 mrClscmrc 17534 mrIndcmri 17535 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-mre 17537 df-mrc 17538 df-mri 17539 |
This theorem is referenced by: (None) |
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