Theorem ssrexr 41713

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 10687	. 2 ⊢ ℝ ⊆ ℝ*
3	1, 2	sstrdi 3981	1 ⊢ (𝜑 → 𝐴 ⊆ ℝ*)

Syntax hints:   → wi 4   ⊆ wss 3938  ℝcr 10538  ℝ^*cxr 10676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-un 3943  df-in 3945  df-ss 3954  df-xr 10681

This theorem is referenced by:  limsuppnfdlem  41989  liminfval2  42056