Theorem mrieqvd 17579

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 17574	. 2 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
6	3	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ (Moore‘𝑋))
7	4	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ 𝑋)
8		simpr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
9	6, 1, 7, 8	mrieqvlemd 17570	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁‘𝑆)))
10	9	necon3bbid 2979	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁‘𝑆)))
11	10	ralbidva 3176	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑆 (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁‘𝑆)))
12	5, 11	bitrd 279	1 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁‘𝑆)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062   ∖ cdif 3945   ⊆ wss 3948  {csn 4628  ‘cfv 6541  Moorecmre 17523  mrClscmrc 17524  mrIndcmri 17525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-mre 17527  df-mrc 17528  df-mri 17529

This theorem is referenced by: (None)