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Metamath Proof Explorer

Theorem uffinfix

Description: An ultrafilter containing a finite element is fixed. (Contributed by Jeff Hankins, 5-Dec-2009) (Revised by Stefan O'Rear, 2-Aug-2015)

		Ref	Expression
	Assertion	uffinfix	$⊢ (F \in UFil (X) \land S \in F \land S \in Fin) \to \exists x \in X F = \{y \in 𝒫 X \| x \in y\}$

Proof

Step	Hyp	Ref	Expression
1		ufilfil	$⊢ F \in UFil (X) \to F \in Fil (X)$
2		filfinnfr	$⊢ (F \in Fil (X) \land S \in F \land S \in Fin) \to ⋂ F \neq \emptyset$
3	1 2	syl3an1	$⊢ (F \in UFil (X) \land S \in F \land S \in Fin) \to ⋂ F \neq \emptyset$
4		uffix2	$⊢ F \in UFil (X) \to (⋂ F \neq \emptyset \leftrightarrow \exists x \in X F = \{y \in 𝒫 X \| x \in y\})$
5	4	3ad2ant1	$⊢ (F \in UFil (X) \land S \in F \land S \in Fin) \to (⋂ F \neq \emptyset \leftrightarrow \exists x \in X F = \{y \in 𝒫 X \| x \in y\})$
6	3 5	mpbid	$⊢ (F \in UFil (X) \land S \in F \land S \in Fin) \to \exists x \in X F = \{y \in 𝒫 X \| x \in y\}$