Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ismred | Structured version Visualization version GIF version |
Description: Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
ismred.ss | β’ (π β πΆ β π« π) |
ismred.ba | β’ (π β π β πΆ) |
ismred.in | β’ ((π β§ π β πΆ β§ π β β ) β β© π β πΆ) |
Ref | Expression |
---|---|
ismred | β’ (π β πΆ β (Mooreβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismred.ss | . 2 β’ (π β πΆ β π« π) | |
2 | ismred.ba | . 2 β’ (π β π β πΆ) | |
3 | velpw 4564 | . . . 4 β’ (π β π« πΆ β π β πΆ) | |
4 | ismred.in | . . . . 5 β’ ((π β§ π β πΆ β§ π β β ) β β© π β πΆ) | |
5 | 4 | 3expia 1122 | . . . 4 β’ ((π β§ π β πΆ) β (π β β β β© π β πΆ)) |
6 | 3, 5 | sylan2b 595 | . . 3 β’ ((π β§ π β π« πΆ) β (π β β β β© π β πΆ)) |
7 | 6 | ralrimiva 3142 | . 2 β’ (π β βπ β π« πΆ(π β β β β© π β πΆ)) |
8 | ismre 17405 | . 2 β’ (πΆ β (Mooreβπ) β (πΆ β π« π β§ π β πΆ β§ βπ β π« πΆ(π β β β β© π β πΆ))) | |
9 | 1, 2, 7, 8 | syl3anbrc 1344 | 1 β’ (π β πΆ β (Mooreβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 β wcel 2107 β wne 2942 βwral 3063 β wss 3909 β c0 4281 π« cpw 4559 β© cint 4906 βcfv 6492 Moorecmre 17397 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6444 df-fun 6494 df-fv 6500 df-mre 17401 |
This theorem is referenced by: ismred2 17418 mremre 17419 submre 17420 subrgmre 20170 lssmre 20351 cssmre 21021 cldmre 22352 toponmre 22367 zartopn 32230 ismrcd1 40887 |
Copyright terms: Public domain | W3C validator |