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	$$⊢\left(M\in {\mathbb{N}}_0\wedge R\in V\wedge F:R⟶{\mathbb{N}}_0\right)\to M\mathrm{Ramsey}F\in {ℝ}^{*}$$

Proof

Step	Hyp	Ref	Expression
1		nn0ssre	$$⊢{\mathbb{N}}_0\subseteq ℝ$$
2		ressxr	$$⊢ℝ\subseteq {ℝ}^{*}$$
3	1 2	sstri	$$⊢{\mathbb{N}}_0\subseteq {ℝ}^{*}$$
4		pnfxr	$$⊢\mathrm{+\infty }\in {ℝ}^{*}$$
5		snssi	$$⊢\mathrm{+\infty }\in {ℝ}^{*}\to \left\{\mathrm{+\infty }\right\}\subseteq {ℝ}^{*}$$
6	4 5	ax-mp	$$⊢\left\{\mathrm{+\infty }\right\}\subseteq {ℝ}^{*}$$
7	3 6	unssi	$$⊢{\mathbb{N}}_0\cup \left\{\mathrm{+\infty }\right\}\subseteq {ℝ}^{*}$$
8		ramcl2	$$⊢\left(M\in {\mathbb{N}}_0\wedge R\in V\wedge F:R⟶{\mathbb{N}}_0\right)\to M\mathrm{Ramsey}F\in \left({\mathbb{N}}_0\cup \left\{\mathrm{+\infty }\right\}\right)$$
9	7 8	sselid	$$⊢\left(M\in {\mathbb{N}}_0\wedge R\in V\wedge F:R⟶{\mathbb{N}}_0\right)\to M\mathrm{Ramsey}F\in {ℝ}^{*}$$