Theorem mrieqvd 17623

Description: In a Moore system, a set is independent if and only if, for all elements of the set, the closure of the set with the element removed is unequal to the closure of the original set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
mrieqvd.1	⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
mrieqvd.2	⊢ 𝑁 = (mrCls‘𝐴)
mrieqvd.3	⊢ 𝐼 = (mrInd‘𝐴)
mrieqvd.4	⊢ (𝜑 → 𝑆 ⊆ 𝑋)

Assertion

Ref	Expression
mrieqvd	⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁‘𝑆)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑆   𝜑,𝑥

Allowed substitution hints:   𝐼(𝑥)   𝑁(𝑥)   𝑋(𝑥)

Proof of Theorem mrieqvd

Step	Hyp	Ref	Expression
1		mrieqvd.2	. . 3 ⊢ 𝑁 = (mrCls‘𝐴)
2		mrieqvd.3	. . 3 ⊢ 𝐼 = (mrInd‘𝐴)
3		mrieqvd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
4		mrieqvd.4	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
5	1, 2, 3, 4	ismri2d 17618	. 2 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
6	3	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ (Moore‘𝑋))
7	4	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ 𝑋)
8		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
9	6, 1, 7, 8	mrieqvlemd 17614	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘𝑆)))
10	9	necon3bbid 2974	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁‘𝑆)))
11	10	ralbidva 3171	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑆 (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁‘𝑆)))
12	5, 11	bitrd 278	1 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁‘𝑆)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2936  ∀wral 3057   ∖ cdif 3944   ⊆ wss 3947  {csn 4630  ‘cfv 6551  Moorecmre 17567  mrClscmrc 17568  mrIndcmri 17569

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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559  df-mre 17571  df-mrc 17572  df-mri 17573

This theorem is referenced by: (None)