Theorem ismri2

Description: Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	ismri2.1	⊢ 𝑁 = ( mrCls ‘ 𝐴 )
	ismri2.2	⊢ 𝐼 = ( mrInd ‘ 𝐴 )
Assertion	ismri2	⊢ ( ( 𝐴 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑆 ∈ 𝐼 ↔ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ismri2.1	⊢ 𝑁 = ( mrCls ‘ 𝐴 )
2		ismri2.2	⊢ 𝐼 = ( mrInd ‘ 𝐴 )
3	1 2	ismri	⊢ ( 𝐴 ∈ ( Moore ‘ 𝑋 ) → ( 𝑆 ∈ 𝐼 ↔ ( 𝑆 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) ) )
4	3	baibd	⊢ ( ( 𝐴 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑆 ∈ 𝐼 ↔ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) )