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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 17504. (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 17504 | . 2 β’ ((π΄ β (Mooreβπ) β§ π β π) β π β (πβπ)) |
5 | 1, 2, 4 | syl2anc 585 | 1 β’ (π β π β (πβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3915 βcfv 6501 Moorecmre 17469 mrClscmrc 17470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-mre 17473 df-mrc 17474 |
This theorem is referenced by: submrc 17515 mrieqvlemd 17516 mrieqv2d 17526 mreexmrid 17530 mreexexlem2d 17532 mreexexlem3d 17533 mreexfidimd 17537 isacs2 17540 acsmap2d 18451 cycsubg2cl 19011 odf1o1 19361 gsumzsplit 19711 gsumzoppg 19728 gsumpt 19746 dprdfeq0 19808 dprdspan 19813 subgdmdprd 19820 subgdprd 19821 dprd2dlem1 19827 dprd2da 19828 dmdprdsplit2lem 19831 pgpfac1lem1 19860 pgpfac1lem3a 19862 pgpfac1lem3 19863 pgpfac1lem5 19865 pgpfaclem2 19868 proot1mul 41555 |
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