Theorem ssrexr 46179

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 11264	. 2 ⊢ ℝ ⊆ ℝ*
3	1, 2	sstrdi 3950	1 ⊢ (𝜑 → 𝐴 ⊆ ℝ*)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3906  ℝcr 11110  ℝ^*cxr 11253

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3911  df-ss 3923  df-xr 11258

This theorem is used by:  limsuppnfdlem  46448  limsupvaluz2  46485  liminfval2  46515