Theorem ssrexr 46129

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 11254	. 2 ⊢ ℝ ⊆ ℝ*
3	1, 2	sstrdi 3950	1 ⊢ (𝜑 → 𝐴 ⊆ ℝ*)

Syntax hints:   → wi 4   ⊆ wss 3906  ℝcr 11100  ℝ^*cxr 11243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-un 3911  df-ss 3923  df-xr 11248

This theorem is referenced by:  limsuppnfdlem  46398  limsupvaluz2  46435  liminfval2  46465