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Mirrors > Home > MPE Home > Th. List > mrieqvd | Structured version Visualization version Unicode version |
Description: In a Moore system, a set is independent if and only if, for all elements of the set, the closure of the set with the element removed is unequal to the closure of the original set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mrieqvd.1 |
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mrieqvd.2 |
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mrieqvd.3 |
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mrieqvd.4 |
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Ref | Expression |
---|---|
mrieqvd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrieqvd.2 |
. . 3
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2 | mrieqvd.3 |
. . 3
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3 | mrieqvd.1 |
. . 3
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4 | mrieqvd.4 |
. . 3
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5 | 1, 2, 3, 4 | ismri2d 15539 |
. 2
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6 | 3 | adantr 467 |
. . . . 5
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7 | 4 | adantr 467 |
. . . . 5
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8 | simpr 463 |
. . . . 5
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9 | 6, 1, 7, 8 | mrieqvlemd 15535 |
. . . 4
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10 | 9 | necon3bbid 2661 |
. . 3
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11 | 10 | ralbidva 2824 |
. 2
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12 | 5, 11 | bitrd 257 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-int 4235 df-br 4403 df-opab 4462 df-mpt 4463 df-id 4749 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-fv 5590 df-mre 15492 df-mrc 15493 df-mri 15494 |
This theorem is referenced by: (None) |
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