Theorem ismri2dad 17327

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	⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
ismri2dad.4	⊢ (𝜑 → 𝑆 ∈ 𝐼)
ismri2dad.5	⊢ (𝜑 → 𝑌 ∈ 𝑆)

Assertion

Ref	Expression
ismri2dad	⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))

Proof of Theorem ismri2dad

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismri2dad.4	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐼)
2		ismri2dad.1	. . . 4 ⊢ 𝑁 = (mrCls‘𝐴)
3		ismri2dad.2	. . . 4 ⊢ 𝐼 = (mrInd‘𝐴)
4		ismri2dad.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
5	3, 4, 1	mrissd 17326	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
6	2, 3, 4, 5	ismri2d 17323	. . 3 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
7	1, 6	mpbid 231	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
8		ismri2dad.5	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
9		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → 𝑥 = 𝑌)
10	9	sneqd 4578	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → {𝑥} = {𝑌})
11	10	difeq2d 4061	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑌}))
12	11	fveq2d 6772	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑌})))
13	9, 12	eleq12d 2834	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
14	13	notbid 317	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
15	8, 14	rspcdv 3551	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
16	7, 15	mpd 15	1 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109  ∀wral 3065   ∖ cdif 3888  {csn 4566  ‘cfv 6430  Moorecmre 17272  mrClscmrc 17273  mrIndcmri 17274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-iota 6388  df-fun 6432  df-fv 6438  df-mre 17276  df-mri 17278

This theorem is referenced by:  mrieqv2d  17329  mreexmrid  17333  mreexexlem2d  17335  acsfiindd  18252