Theorem ssrexr 42972

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 11019	. 2 ⊢ ℝ ⊆ ℝ*
3	1, 2	sstrdi 3933	1 ⊢ (𝜑 → 𝐴 ⊆ ℝ*)

Syntax hints:   → wi 4   ⊆ wss 3887  ℝcr 10870  ℝ^*cxr 11008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-in 3894  df-ss 3904  df-xr 11013

This theorem is referenced by:  limsuppnfdlem  43242  liminfval2  43309