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 ∈ ( UFil ‘ 𝑋 ) ∧ 𝑆 ∈ 𝐹 ∧ 𝑆 ∈ Fin ) → ∃ 𝑥 ∈ 𝑋 𝐹 = { 𝑦 ∈ 𝒫 𝑋 ∣ 𝑥 ∈ 𝑦 } )

Proof

Step	Hyp	Ref	Expression
1		ufilfil	⊢ ( 𝐹 ∈ ( UFil ‘ 𝑋 ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) )
2		filfinnfr	⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑆 ∈ 𝐹 ∧ 𝑆 ∈ Fin ) → ∩ 𝐹 ≠ ∅ )
3	1 2	syl3an1	⊢ ( ( 𝐹 ∈ ( UFil ‘ 𝑋 ) ∧ 𝑆 ∈ 𝐹 ∧ 𝑆 ∈ Fin ) → ∩ 𝐹 ≠ ∅ )
4		uffix2	⊢ ( 𝐹 ∈ ( UFil ‘ 𝑋 ) → ( ∩ 𝐹 ≠ ∅ ↔ ∃ 𝑥 ∈ 𝑋 𝐹 = { 𝑦 ∈ 𝒫 𝑋 ∣ 𝑥 ∈ 𝑦 } ) )
5	4	3ad2ant1	⊢ ( ( 𝐹 ∈ ( UFil ‘ 𝑋 ) ∧ 𝑆 ∈ 𝐹 ∧ 𝑆 ∈ Fin ) → ( ∩ 𝐹 ≠ ∅ ↔ ∃ 𝑥 ∈ 𝑋 𝐹 = { 𝑦 ∈ 𝒫 𝑋 ∣ 𝑥 ∈ 𝑦 } ) )
6	3 5	mpbid	⊢ ( ( 𝐹 ∈ ( UFil ‘ 𝑋 ) ∧ 𝑆 ∈ 𝐹 ∧ 𝑆 ∈ Fin ) → ∃ 𝑥 ∈ 𝑋 𝐹 = { 𝑦 ∈ 𝒫 𝑋 ∣ 𝑥 ∈ 𝑦 } )