Theorem rexpssxrxp 11357

Description: The Cartesian product of standard reals are a subset of the Cartesian product of extended reals. (Contributed by David A. Wheeler, 8-Dec-2018.)

Assertion

Ref	Expression
rexpssxrxp	⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)

Proof of Theorem rexpssxrxp

Step	Hyp	Ref	Expression
1		ressxr 11356	. 2 ⊢ ℝ ⊆ ℝ*
2		xpss12 5665	. 2 ⊢ ((ℝ ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
3	1, 1, 2	mp2an 705	1 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)

This proof depends on syntax axioms:   ⊆ wss 3895   × cxp 5648  ℝcr 11202  ℝ^*cxr 11345

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-un 4151  df-opab 5167  df-xp 5656  df-xr 11350

This theorem is used by:  ltrelxr  11373  xrsdsre  25135  ovolfioo  25793  ovolficc  25794  ovolficcss  25795  ovollb  25805  ovolicc2  25848  ovolfs2  25897  uniiccdif  25904  uniioovol  25905  uniiccvol  25906  uniioombllem2  25909  uniioombllem3a  25910  uniioombllem3  25911  uniioombllem4  25912  uniioombllem5  25913  uniioombl  25915  dyadmbllem  25925  opnmbllem  25927  icoreresf  38368  icoreelrn  38377  relowlpssretop  38380  opnmbllem0  38667  mblfinlem1  38668  mblfinlem2  38669  voliooicof  47099  ovolval3  47750  ovolval4lem2  47753  ovolval5lem2  47756  ovolval5lem3  47757  ovnovollem1  47759  ovnovollem2  47760