Theorem rexpssxrxp 11280

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 11279	. 2 ⊢ ℝ ⊆ ℝ*
2		xpss12 5669	. 2 ⊢ ((ℝ ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
3	1, 1, 2	mp2an 692	1 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)

Syntax hints:   ⊆ wss 3926   × cxp 5652  ℝcr 11128  ℝ^*cxr 11268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931  df-ss 3943  df-opab 5182  df-xp 5660  df-xr 11273

This theorem is referenced by:  ltrelxr  11296  xrsdsre  24750  ovolfioo  25420  ovolficc  25421  ovolficcss  25422  ovollb  25432  ovolicc2  25475  ovolfs2  25524  uniiccdif  25531  uniioovol  25532  uniiccvol  25533  uniioombllem2  25536  uniioombllem3a  25537  uniioombllem3  25538  uniioombllem4  25539  uniioombllem5  25540  uniioombl  25542  dyadmbllem  25552  opnmbllem  25554  icoreresf  37370  icoreelrn  37379  relowlpssretop  37382  opnmbllem0  37680  mblfinlem1  37681  mblfinlem2  37682  voliooicof  46025  ovolval3  46676  ovolval4lem2  46679  ovolval5lem2  46682  ovolval5lem3  46683  ovnovollem1  46685  ovnovollem2  46686