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Theorem ismri2dad 17164
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 = (mrCls‘𝐴)
ismri2dad.2 𝐼 = (mrInd‘𝐴)
ismri2dad.3 (𝜑𝐴 ∈ (Moore‘𝑋))
ismri2dad.4 (𝜑𝑆𝐼)
ismri2dad.5 (𝜑𝑌𝑆)
Assertion
Ref Expression
ismri2dad (𝜑 → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))

Proof of Theorem ismri2dad
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ismri2dad.4 . . 3 (𝜑𝑆𝐼)
2 ismri2dad.1 . . . 4 𝑁 = (mrCls‘𝐴)
3 ismri2dad.2 . . . 4 𝐼 = (mrInd‘𝐴)
4 ismri2dad.3 . . . 4 (𝜑𝐴 ∈ (Moore‘𝑋))
53, 4, 1mrissd 17163 . . . 4 (𝜑𝑆𝑋)
62, 3, 4, 5ismri2d 17160 . . 3 (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
71, 6mpbid 235 . 2 (𝜑 → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
8 ismri2dad.5 . . 3 (𝜑𝑌𝑆)
9 simpr 488 . . . . 5 ((𝜑𝑥 = 𝑌) → 𝑥 = 𝑌)
109sneqd 4567 . . . . . . 7 ((𝜑𝑥 = 𝑌) → {𝑥} = {𝑌})
1110difeq2d 4051 . . . . . 6 ((𝜑𝑥 = 𝑌) → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑌}))
1211fveq2d 6739 . . . . 5 ((𝜑𝑥 = 𝑌) → (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑌})))
139, 12eleq12d 2833 . . . 4 ((𝜑𝑥 = 𝑌) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
1413notbid 321 . . 3 ((𝜑𝑥 = 𝑌) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
158, 14rspcdv 3541 . 2 (𝜑 → (∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
167, 15mpd 15 1 (𝜑 → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1543  wcel 2111  wral 3062  cdif 3877  {csn 4555  cfv 6397  Moorecmre 17109  mrClscmrc 17110  mrIndcmri 17111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pow 5272  ax-pr 5336  ax-un 7541
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-pw 4529  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-opab 5130  df-mpt 5150  df-id 5469  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-rn 5576  df-iota 6355  df-fun 6399  df-fv 6405  df-mre 17113  df-mri 17115
This theorem is referenced by:  mrieqv2d  17166  mreexmrid  17170  mreexexlem2d  17172  acsfiindd  18083
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