Theorem rexrd 7535

Description: A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)

Hypothesis

Ref	Expression
rexrd.1	|- ( ph -> A e. RR )

Assertion

Ref	Expression
rexrd	|- ( ph -> A e. RR* )

Proof of Theorem rexrd

Step	Hyp	Ref	Expression
1		ressxr 7529	. 2 |- RR C_ RR*
2		rexrd.1	. 2 |- ( ph -> A e. RR )
3	1, 2	sseldi 3023	1 |- ( ph -> A e. RR* )

Syntax hints:   -> wi 4   e. wcel 1438   RRcr 7347   RR*cxr 7519

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-xr 7524

This theorem is referenced by:  xnn0xr  8739  rpxr  9139  rpxrd  9172  xnegcl  9292  iooshf  9368  icoshftf1o  9406  ioo0  9667  ioom  9668  ico0  9669  ioc0  9670  modqelico  9737  mulqaddmodid  9767  addmodid  9775  elicc4abs  10523