Theorem ismri2dad 17603

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 17602	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
6	2, 3, 4, 5	ismri2d 17599	. . 3 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
7	1, 6	mpbid 232	. 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 4579	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → {𝑥} = {𝑌})
11	10	difeq2d 4066	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑌}))
12	11	fveq2d 6844	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑌})))
13	9, 12	eleq12d 2830	. . . 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 3556	. 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 1542   ∈ wcel 2114  ∀wral 3051   ∖ cdif 3886  {csn 4567  ‘cfv 6498  Moorecmre 17544  mrClscmrc 17545  mrIndcmri 17546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fv 6506  df-mre 17548  df-mri 17550

This theorem is referenced by:  mrieqv2d  17605  mreexmrid  17609  mreexexlem2d  17611  acsfiindd  18519