Theorem ssrexr 45428

Description: A subset of the reals is a subset of the extended reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
ssrexr.1	⊢ (𝜑 → 𝐴 ⊆ ℝ)

Assertion

Ref	Expression
ssrexr	⊢ (𝜑 → 𝐴 ⊆ ℝ*)

Proof of Theorem ssrexr

Step	Hyp	Ref	Expression
1		ssrexr.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
2		ressxr 11218	. 2 ⊢ ℝ ⊆ ℝ*
3	1, 2	sstrdi 3959	1 ⊢ (𝜑 → 𝐴 ⊆ ℝ*)

Syntax hints:   → wi 4   ⊆ wss 3914  ℝcr 11067  ℝ^*cxr 11207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-un 3919  df-ss 3931  df-xr 11212

This theorem is referenced by:  limsuppnfdlem  45699  limsupvaluz2  45736  liminfval2  45766