Description: In a Moore system, if an independent set is between a set and its closure, the two sets are equal (since the two sets must have equal closures by mressmrcd , and so are equal by mrieqv2d .) (Contributed by David Moews, 1-May-2017)
Ref | Expression | ||
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Hypotheses | mrissmrcd.1 | |
|
mrissmrcd.2 | |
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mrissmrcd.3 | |
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mrissmrcd.4 | |
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mrissmrcd.5 | |
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mrissmrcd.6 | |
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Assertion | mrissmrcd | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrissmrcd.1 | |
|
2 | mrissmrcd.2 | |
|
3 | mrissmrcd.3 | |
|
4 | mrissmrcd.4 | |
|
5 | mrissmrcd.5 | |
|
6 | mrissmrcd.6 | |
|
7 | 1 2 4 5 | mressmrcd | |
8 | pssne | |
|
9 | 8 | necomd | |
10 | 9 | necon2bi | |
11 | 7 10 | syl | |
12 | 3 1 6 | mrissd | |
13 | 1 2 3 12 | mrieqv2d | |
14 | 6 13 | mpbid | |
15 | 6 5 | ssexd | |
16 | simpr | |
|
17 | 16 | psseq1d | |
18 | 16 | fveq2d | |
19 | 18 | psseq1d | |
20 | 17 19 | imbi12d | |
21 | 15 20 | spcdv | |
22 | 14 21 | mpd | |
23 | 11 22 | mtod | |
24 | sspss | |
|
25 | 5 24 | sylib | |
26 | 25 | ord | |
27 | 23 26 | mpd | |
28 | 27 | eqcomd | |