Theorem ramxrcl

Description: The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl .) (Contributed by Mario Carneiro, 20-Apr-2015)

	Ref	Expression
Assertion	ramxrcl	⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹 : 𝑅 ⟶ ℕ0 ) → ( 𝑀 Ramsey 𝐹 ) ∈ ℝ* )

Proof

Step	Hyp	Ref	Expression
1		nn0ssre	⊢ ℕ0 ⊆ ℝ
2		ressxr	⊢ ℝ ⊆ ℝ*
3	1 2	sstri	⊢ ℕ0 ⊆ ℝ*
4		pnfxr	⊢ +∞ ∈ ℝ*
5		snssi	⊢ ( +∞ ∈ ℝ* → { +∞ } ⊆ ℝ* )
6	4 5	ax-mp	⊢ { +∞ } ⊆ ℝ*
7	3 6	unssi	⊢ ( ℕ0 ∪ { +∞ } ) ⊆ ℝ*
8		ramcl2	⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹 : 𝑅 ⟶ ℕ0 ) → ( 𝑀 Ramsey 𝐹 ) ∈ ( ℕ0 ∪ { +∞ } ) )
9	7 8	sseldi	⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹 : 𝑅 ⟶ ℕ0 ) → ( 𝑀 Ramsey 𝐹 ) ∈ ℝ* )