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Mirrors > Home > MPE Home > Th. List > ismri2dad | Structured version Visualization version GIF version |
Description: Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
ismri2dad.1 | β’ π = (mrClsβπ΄) |
ismri2dad.2 | β’ πΌ = (mrIndβπ΄) |
ismri2dad.3 | β’ (π β π΄ β (Mooreβπ)) |
ismri2dad.4 | β’ (π β π β πΌ) |
ismri2dad.5 | β’ (π β π β π) |
Ref | Expression |
---|---|
ismri2dad | β’ (π β Β¬ π β (πβ(π β {π}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismri2dad.4 | . . 3 β’ (π β π β πΌ) | |
2 | ismri2dad.1 | . . . 4 β’ π = (mrClsβπ΄) | |
3 | ismri2dad.2 | . . . 4 β’ πΌ = (mrIndβπ΄) | |
4 | ismri2dad.3 | . . . 4 β’ (π β π΄ β (Mooreβπ)) | |
5 | 3, 4, 1 | mrissd 17442 | . . . 4 β’ (π β π β π) |
6 | 2, 3, 4, 5 | ismri2d 17439 | . . 3 β’ (π β (π β πΌ β βπ₯ β π Β¬ π₯ β (πβ(π β {π₯})))) |
7 | 1, 6 | mpbid 231 | . 2 β’ (π β βπ₯ β π Β¬ π₯ β (πβ(π β {π₯}))) |
8 | ismri2dad.5 | . . 3 β’ (π β π β π) | |
9 | simpr 485 | . . . . 5 β’ ((π β§ π₯ = π) β π₯ = π) | |
10 | 9 | sneqd 4585 | . . . . . . 7 β’ ((π β§ π₯ = π) β {π₯} = {π}) |
11 | 10 | difeq2d 4069 | . . . . . 6 β’ ((π β§ π₯ = π) β (π β {π₯}) = (π β {π})) |
12 | 11 | fveq2d 6829 | . . . . 5 β’ ((π β§ π₯ = π) β (πβ(π β {π₯})) = (πβ(π β {π}))) |
13 | 9, 12 | eleq12d 2831 | . . . 4 β’ ((π β§ π₯ = π) β (π₯ β (πβ(π β {π₯})) β π β (πβ(π β {π})))) |
14 | 13 | notbid 317 | . . 3 β’ ((π β§ π₯ = π) β (Β¬ π₯ β (πβ(π β {π₯})) β Β¬ π β (πβ(π β {π})))) |
15 | 8, 14 | rspcdv 3562 | . 2 β’ (π β (βπ₯ β π Β¬ π₯ β (πβ(π β {π₯})) β Β¬ π β (πβ(π β {π})))) |
16 | 7, 15 | mpd 15 | 1 β’ (π β Β¬ π β (πβ(π β {π}))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 = wceq 1540 β wcel 2105 βwral 3061 β cdif 3895 {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: mrieqv2d 17445 mreexmrid 17449 mreexexlem2d 17451 acsfiindd 18368 |
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