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Mirrors > Home > MPE Home > Th. List > mrcssidd | Structured version Visualization version GIF version |
Description: A set is contained in its Moore closure. Deduction form of mrcssid 17560. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mrcssidd.1 | β’ (π β π΄ β (Mooreβπ)) |
mrcssidd.2 | β’ π = (mrClsβπ΄) |
mrcssidd.3 | β’ (π β π β π) |
Ref | Expression |
---|---|
mrcssidd | β’ (π β π β (πβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrcssidd.1 | . 2 β’ (π β π΄ β (Mooreβπ)) | |
2 | mrcssidd.3 | . 2 β’ (π β π β π) | |
3 | mrcssidd.2 | . . 3 β’ π = (mrClsβπ΄) | |
4 | 3 | mrcssid 17560 | . 2 β’ ((π΄ β (Mooreβπ) β§ π β π) β π β (πβπ)) |
5 | 1, 2, 4 | syl2anc 584 | 1 β’ (π β π β (πβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 β wss 3948 βcfv 6543 Moorecmre 17525 mrClscmrc 17526 |
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 |
This theorem is referenced by: submrc 17571 mrieqvlemd 17572 mrieqv2d 17582 mreexmrid 17586 mreexexlem2d 17588 mreexexlem3d 17589 mreexfidimd 17593 isacs2 17596 acsmap2d 18507 cycsubg2cl 19087 odf1o1 19439 gsumzsplit 19794 gsumzoppg 19811 gsumpt 19829 dprdfeq0 19891 dprdspan 19896 subgdmdprd 19903 subgdprd 19904 dprd2dlem1 19910 dprd2da 19911 dmdprdsplit2lem 19914 pgpfac1lem1 19943 pgpfac1lem3a 19945 pgpfac1lem3 19946 pgpfac1lem5 19948 pgpfaclem2 19951 proot1mul 41931 |
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