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Theorem ismri2dad 17616
Description: Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ismri2dad.1 𝑁 = (mrClsβ€˜π΄)
ismri2dad.2 𝐼 = (mrIndβ€˜π΄)
ismri2dad.3 (πœ‘ β†’ 𝐴 ∈ (Mooreβ€˜π‘‹))
ismri2dad.4 (πœ‘ β†’ 𝑆 ∈ 𝐼)
ismri2dad.5 (πœ‘ β†’ π‘Œ ∈ 𝑆)
Assertion
Ref Expression
ismri2dad (πœ‘ β†’ Β¬ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ})))

Proof of Theorem ismri2dad
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 ismri2dad.4 . . 3 (πœ‘ β†’ 𝑆 ∈ 𝐼)
2 ismri2dad.1 . . . 4 𝑁 = (mrClsβ€˜π΄)
3 ismri2dad.2 . . . 4 𝐼 = (mrIndβ€˜π΄)
4 ismri2dad.3 . . . 4 (πœ‘ β†’ 𝐴 ∈ (Mooreβ€˜π‘‹))
53, 4, 1mrissd 17615 . . . 4 (πœ‘ β†’ 𝑆 βŠ† 𝑋)
62, 3, 4, 5ismri2d 17612 . . 3 (πœ‘ β†’ (𝑆 ∈ 𝐼 ↔ βˆ€π‘₯ ∈ 𝑆 Β¬ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯}))))
71, 6mpbid 231 . 2 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑆 Β¬ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯})))
8 ismri2dad.5 . . 3 (πœ‘ β†’ π‘Œ ∈ 𝑆)
9 simpr 483 . . . . 5 ((πœ‘ ∧ π‘₯ = π‘Œ) β†’ π‘₯ = π‘Œ)
109sneqd 4636 . . . . . . 7 ((πœ‘ ∧ π‘₯ = π‘Œ) β†’ {π‘₯} = {π‘Œ})
1110difeq2d 4114 . . . . . 6 ((πœ‘ ∧ π‘₯ = π‘Œ) β†’ (𝑆 βˆ– {π‘₯}) = (𝑆 βˆ– {π‘Œ}))
1211fveq2d 6896 . . . . 5 ((πœ‘ ∧ π‘₯ = π‘Œ) β†’ (π‘β€˜(𝑆 βˆ– {π‘₯})) = (π‘β€˜(𝑆 βˆ– {π‘Œ})))
139, 12eleq12d 2819 . . . 4 ((πœ‘ ∧ π‘₯ = π‘Œ) β†’ (π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯})) ↔ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ}))))
1413notbid 317 . . 3 ((πœ‘ ∧ π‘₯ = π‘Œ) β†’ (Β¬ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯})) ↔ Β¬ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ}))))
158, 14rspcdv 3593 . 2 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝑆 Β¬ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯})) β†’ Β¬ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ}))))
167, 15mpd 15 1 (πœ‘ β†’ Β¬ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ})))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051   βˆ– cdif 3936  {csn 4624  β€˜cfv 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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