Theorem ssrexr 39659

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

Hypothesis

Ref	Expression
ssrexr.1	|- ( ph -> A C_ RR )

Assertion

Ref	Expression
ssrexr	|- ( ph -> A C_ RR* )

Proof of Theorem ssrexr

Step	Hyp	Ref	Expression
1		ssrexr.1	. 2 |- ( ph -> A C_ RR )
2		ressxr 10083	. . 3 |- RR C_ RR*
3	2	a1i 11	. 2 |- ( ph -> RR C_ RR* )
4	1, 3	sstrd 3613	1 |- ( ph -> A C_ RR* )

Syntax hints:   -> wi 4   C_ wss 3574   RRcr 9935   RR*cxr 10073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-xr 10078

This theorem is referenced by:  liminfval2  40000