![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 17565. (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 17565 | . 2 β’ ((π΄ β (Mooreβπ) β§ π β π) β π β (πβπ)) |
5 | 1, 2, 4 | syl2anc 582 | 1 β’ (π β π β (πβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 β wss 3947 βcfv 6542 Moorecmre 17530 mrClscmrc 17531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-mre 17534 df-mrc 17535 |
This theorem is referenced by: submrc 17576 mrieqvlemd 17577 mrieqv2d 17587 mreexmrid 17591 mreexexlem2d 17593 mreexexlem3d 17594 mreexfidimd 17598 isacs2 17601 acsmap2d 18512 cycsubg2cl 19126 odf1o1 19481 gsumzsplit 19836 gsumzoppg 19853 gsumpt 19871 dprdfeq0 19933 dprdspan 19938 subgdmdprd 19945 subgdprd 19946 dprd2dlem1 19952 dprd2da 19953 dmdprdsplit2lem 19956 pgpfac1lem1 19985 pgpfac1lem3a 19987 pgpfac1lem3 19988 pgpfac1lem5 19990 pgpfaclem2 19993 proot1mul 42243 |
Copyright terms: Public domain | W3C validator |