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Mirrors > Home > MPE Home > Th. List > ismri2d | Structured version Visualization version GIF version |
Description: Criterion for a subset of the base set in a Moore system to be independent. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
ismri2.1 | β’ π = (mrClsβπ΄) |
ismri2.2 | β’ πΌ = (mrIndβπ΄) |
ismri2d.3 | β’ (π β π΄ β (Mooreβπ)) |
ismri2d.4 | β’ (π β π β π) |
Ref | Expression |
---|---|
ismri2d | β’ (π β (π β πΌ β βπ₯ β π Β¬ π₯ β (πβ(π β {π₯})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismri2d.3 | . 2 β’ (π β π΄ β (Mooreβπ)) | |
2 | ismri2d.4 | . 2 β’ (π β π β π) | |
3 | ismri2.1 | . . 3 β’ π = (mrClsβπ΄) | |
4 | ismri2.2 | . . 3 β’ πΌ = (mrIndβπ΄) | |
5 | 3, 4 | ismri2 17438 | . 2 β’ ((π΄ β (Mooreβπ) β§ π β π) β (π β πΌ β βπ₯ β π Β¬ π₯ β (πβ(π β {π₯})))) |
6 | 1, 2, 5 | syl2anc 584 | 1 β’ (π β (π β πΌ β βπ₯ β π Β¬ π₯ β (πβ(π β {π₯})))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 = wceq 1540 β wcel 2105 βwral 3061 β cdif 3895 β wss 3898 {csn 4573 βcfv 6479 Moorecmre 17388 mrClscmrc 17389 mrIndcmri 17390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-iota 6431 df-fun 6481 df-fv 6487 df-mre 17392 df-mri 17394 |
This theorem is referenced by: ismri2dd 17440 ismri2dad 17443 mrieqvd 17444 mrieqv2d 17445 mrissmrid 17447 |
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