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Mirrors > Home > MPE Home > Th. List > mrissmrid | Structured version Visualization version GIF version |
Description: In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mrissmrid.1 | β’ (π β π΄ β (Mooreβπ)) |
mrissmrid.2 | β’ π = (mrClsβπ΄) |
mrissmrid.3 | β’ πΌ = (mrIndβπ΄) |
mrissmrid.4 | β’ (π β π β πΌ) |
mrissmrid.5 | β’ (π β π β π) |
Ref | Expression |
---|---|
mrissmrid | β’ (π β π β πΌ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrissmrid.2 | . 2 β’ π = (mrClsβπ΄) | |
2 | mrissmrid.3 | . 2 β’ πΌ = (mrIndβπ΄) | |
3 | mrissmrid.1 | . 2 β’ (π β π΄ β (Mooreβπ)) | |
4 | mrissmrid.5 | . . 3 β’ (π β π β π) | |
5 | mrissmrid.4 | . . . 4 β’ (π β π β πΌ) | |
6 | 2, 3, 5 | mrissd 17615 | . . 3 β’ (π β π β π) |
7 | 4, 6 | sstrd 3983 | . 2 β’ (π β π β π) |
8 | 1, 2, 3, 6 | ismri2d 17612 | . . . 4 β’ (π β (π β πΌ β βπ₯ β π Β¬ π₯ β (πβ(π β {π₯})))) |
9 | 5, 8 | mpbid 231 | . . 3 β’ (π β βπ₯ β π Β¬ π₯ β (πβ(π β {π₯}))) |
10 | 4 | sseld 3971 | . . . . 5 β’ (π β (π₯ β π β π₯ β π)) |
11 | 4 | ssdifd 4133 | . . . . . . 7 β’ (π β (π β {π₯}) β (π β {π₯})) |
12 | 6 | ssdifssd 4135 | . . . . . . 7 β’ (π β (π β {π₯}) β π) |
13 | 3, 1, 11, 12 | mrcssd 17603 | . . . . . 6 β’ (π β (πβ(π β {π₯})) β (πβ(π β {π₯}))) |
14 | 13 | ssneld 3974 | . . . . 5 β’ (π β (Β¬ π₯ β (πβ(π β {π₯})) β Β¬ π₯ β (πβ(π β {π₯})))) |
15 | 10, 14 | imim12d 81 | . . . 4 β’ (π β ((π₯ β π β Β¬ π₯ β (πβ(π β {π₯}))) β (π₯ β π β Β¬ π₯ β (πβ(π β {π₯}))))) |
16 | 15 | ralimdv2 3153 | . . 3 β’ (π β (βπ₯ β π Β¬ π₯ β (πβ(π β {π₯})) β βπ₯ β π Β¬ π₯ β (πβ(π β {π₯})))) |
17 | 9, 16 | mpd 15 | . 2 β’ (π β βπ₯ β π Β¬ π₯ β (πβ(π β {π₯}))) |
18 | 1, 2, 3, 7, 17 | ismri2dd 17613 | 1 β’ (π β π β πΌ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 = wceq 1533 β wcel 2098 βwral 3051 β cdif 3936 β wss 3939 {csn 4624 βcfv 6543 Moorecmre 17561 mrClscmrc 17562 mrIndcmri 17563 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-mre 17565 df-mrc 17566 df-mri 17567 |
This theorem is referenced by: mreexexlem2d 17624 acsfiindd 18544 |
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