Theorem ismri2dad 17695

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 17694	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
6	2, 3, 4, 5	ismri2d 17691	. . 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 4660	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → {𝑥} = {𝑌})
11	10	difeq2d 4149	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑆 ∖ {𝑥}) = (𝑆 ∖ {𝑌}))
12	11	fveq2d 6924	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑌})))
13	9, 12	eleq12d 2838	. . . 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 3627	. 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 1537   ∈ wcel 2108  ∀wral 3067   ∖ cdif 3973  {csn 4648  ‘cfv 6573  Moorecmre 17640  mrClscmrc 17641  mrIndcmri 17642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-mre 17644  df-mri 17646

This theorem is referenced by:  mrieqv2d  17697  mreexmrid  17701  mreexexlem2d  17703  acsfiindd  18623