Theorem ismri2dad 17590

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 17589	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
6	2, 3, 4, 5	ismri2d 17586	. . 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 4635	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → {𝑥} = {𝑌})
11	10	difeq2d 4117	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑌}))
12	11	fveq2d 6889	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑌})))
13	9, 12	eleq12d 2821	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
14	13	notbid 318	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
15	8, 14	rspcdv 3598	. 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 1533   ∈ wcel 2098  ∀wral 3055   ∖ cdif 3940  {csn 4623  ‘cfv 6537  Moorecmre 17535  mrClscmrc 17536  mrIndcmri 17537

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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fv 6545  df-mre 17539  df-mri 17541

This theorem is referenced by:  mrieqv2d  17592  mreexmrid  17596  mreexexlem2d  17598  acsfiindd  18518