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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 17615 | . . . 4 β’ (π β π β π) |
6 | 2, 3, 4, 5 | ismri2d 17612 | . . 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 483 | . . . . 5 β’ ((π β§ π₯ = π) β π₯ = π) | |
10 | 9 | sneqd 4636 | . . . . . . 7 β’ ((π β§ π₯ = π) β {π₯} = {π}) |
11 | 10 | difeq2d 4114 | . . . . . 6 β’ ((π β§ π₯ = π) β (π β {π₯}) = (π β {π})) |
12 | 11 | fveq2d 6896 | . . . . 5 β’ ((π β§ π₯ = π) β (πβ(π β {π₯})) = (πβ(π β {π}))) |
13 | 9, 12 | eleq12d 2819 | . . . 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 3593 | . 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 394 = wceq 1533 β wcel 2098 βwral 3051 β cdif 3936 {csn 4624 βcfv 6543 Moorecmre 17561 mrClscmrc 17562 mrIndcmri 17563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6495 df-fun 6545 df-fv 6551 df-mre 17565 df-mri 17567 |
This theorem is referenced by: mrieqv2d 17618 mreexmrid 17622 mreexexlem2d 17624 acsfiindd 18544 |
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