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Theorem ismri2dad 17443
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 17442 . . . 4 (πœ‘ β†’ 𝑆 βŠ† 𝑋)
62, 3, 4, 5ismri2d 17439 . . 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 485 . . . . 5 ((πœ‘ ∧ π‘₯ = π‘Œ) β†’ π‘₯ = π‘Œ)
109sneqd 4585 . . . . . . 7 ((πœ‘ ∧ π‘₯ = π‘Œ) β†’ {π‘₯} = {π‘Œ})
1110difeq2d 4069 . . . . . 6 ((πœ‘ ∧ π‘₯ = π‘Œ) β†’ (𝑆 βˆ– {π‘₯}) = (𝑆 βˆ– {π‘Œ}))
1211fveq2d 6829 . . . . 5 ((πœ‘ ∧ π‘₯ = π‘Œ) β†’ (π‘β€˜(𝑆 βˆ– {π‘₯})) = (π‘β€˜(𝑆 βˆ– {π‘Œ})))
139, 12eleq12d 2831 . . . 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 3562 . 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 396   = wceq 1540   ∈ wcel 2105  βˆ€wral 3061   βˆ– cdif 3895  {csn 4573  β€˜cfv 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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