Theorem ismri2dad 17595

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 17594	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
6	2, 3, 4, 5	ismri2d 17591	. . 3 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
7	1, 6	mpbid 233	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
8		ismri2dad.5	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
9		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → 𝑥 = 𝑌)
10	9	sneqd 4568	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → {𝑥} = {𝑌})
11	10	difeq2d 4058	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑌}))
12	11	fveq2d 6832	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑌})))
13	9, 12	eleq12d 2833	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
14	13	notbid 319	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))))
15	8, 14	rspcdv 3552	. 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 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ∖ cdif 3880  {csn 4556  ‘cfv 6486  Moorecmre 17536  mrClscmrc 17537  mrIndcmri 17538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fv 6494  df-mre 17540  df-mri 17542

This theorem is referenced by:  mrieqv2d  17597  mreexmrid  17601  mreexexlem2d  17603  acsfiindd  18511