Metamath Proof Explorer

Theorem mreexfidimd

Description: In a Moore system whose closure operator has the exchange property, if two independent sets have equal closure and one is finite, then they are equinumerous. Proven by using mreexdomd twice. This implies a special case of Theorem 4.2.2 in FaureFrolicher p. 87. (Contributed by David Moews, 1-May-2017)

Ref Expression
Hypotheses mreexfidimd.1 ⊢ φ → A ∈ Moore ⁡ X
mreexfidimd.2 ⊢ N = mrCls ⁡ A
mreexfidimd.3 ⊢ I = mrInd ⁡ A
mreexfidimd.4 ⊢ φ → ∀ s ∈ 𝒫 X ∀ y ∈ X ∀ z ∈ N ⁡ s ∪ y ∖ N ⁡ s y ∈ N ⁡ s ∪ z
mreexfidimd.5 ⊢ φ → S ∈ I
mreexfidimd.6 ⊢ φ → T ∈ I
mreexfidimd.7 ⊢ φ → S ∈ Fin
mreexfidimd.8 ⊢ φ → N ⁡ S = N ⁡ T
Assertion mreexfidimd ⊢ φ → S ≈ T

Proof

Step Hyp Ref Expression
1 mreexfidimd.1 ⊢ φ → A ∈ Moore ⁡ X
2 mreexfidimd.2 ⊢ N = mrCls ⁡ A
3 mreexfidimd.3 ⊢ I = mrInd ⁡ A
4 mreexfidimd.4 ⊢ φ → ∀ s ∈ 𝒫 X ∀ y ∈ X ∀ z ∈ N ⁡ s ∪ y ∖ N ⁡ s y ∈ N ⁡ s ∪ z
5 mreexfidimd.5 ⊢ φ → S ∈ I
6 mreexfidimd.6 ⊢ φ → T ∈ I
7 mreexfidimd.7 ⊢ φ → S ∈ Fin
8 mreexfidimd.8 ⊢ φ → N ⁡ S = N ⁡ T
9 3 1 5 mrissd ⊢ φ → S ⊆ X
10 1 2 9 mrcssidd ⊢ φ → S ⊆ N ⁡ S
11 10 8 sseqtrd ⊢ φ → S ⊆ N ⁡ T
12 3 1 6 mrissd ⊢ φ → T ⊆ X
13 7 orcd ⊢ φ → S ∈ Fin ∨ T ∈ Fin
14 1 2 3 4 11 12 13 5 mreexdomd ⊢ φ → S ≼ T
15 1 2 12 mrcssidd ⊢ φ → T ⊆ N ⁡ T
16 15 8 sseqtrrd ⊢ φ → T ⊆ N ⁡ S
17 7 olcd ⊢ φ → T ∈ Fin ∨ S ∈ Fin
18 1 2 3 4 16 9 17 6 mreexdomd ⊢ φ → T ≼ S
19 sbth ⊢ S ≼ T ∧ T ≼ S → S ≈ T
20 14 18 19 syl2anc ⊢ φ → S ≈ T