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Metamath Proof Explorer

Theorem shococss

Description: Inclusion in complement of complement. Part of Proposition 1 of Kalmbach p. 65. (Contributed by NM, 10-Oct-1999) (New usage is discouraged.)

		Ref	Expression
	Assertion	shococss	⊢ ( 𝐴 ∈ S_ℋ → 𝐴 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		shss	⊢ ( 𝐴 ∈ S_ℋ → 𝐴 ⊆ ℋ )
2		ococss	⊢ ( 𝐴 ⊆ ℋ → 𝐴 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) )
3	1 2	syl	⊢ ( 𝐴 ∈ S_ℋ → 𝐴 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) )