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Theorem ismri2dad 16896
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 16895 . . . 4 (𝜑𝑆𝑋)
62, 3, 4, 5ismri2d 16892 . . 3 (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
71, 6mpbid 233 . 2 (𝜑 → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
8 ismri2dad.5 . . 3 (𝜑𝑌𝑆)
9 simpr 485 . . . . 5 ((𝜑𝑥 = 𝑌) → 𝑥 = 𝑌)
109sneqd 4569 . . . . . . 7 ((𝜑𝑥 = 𝑌) → {𝑥} = {𝑌})
1110difeq2d 4096 . . . . . 6 ((𝜑𝑥 = 𝑌) → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑌}))
1211fveq2d 6667 . . . . 5 ((𝜑𝑥 = 𝑌) → (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑌})))
139, 12eleq12d 2904 . . . 4 ((𝜑𝑥 = 𝑌) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
1413notbid 319 . . 3 ((𝜑𝑥 = 𝑌) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
158, 14rspcdv 3612 . 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 396   = wceq 1528  wcel 2105  wral 3135  cdif 3930  {csn 4557  cfv 6348  Moorecmre 16841  mrClscmrc 16842  mrIndcmri 16843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-mre 16845  df-mri 16847
This theorem is referenced by:  mrieqv2d  16898  mreexmrid  16902  mreexexlem2d  16904  acsfiindd  17775
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