Theorem rexpssxrxp 11384

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 11383	. 2 ⊢ ℝ ⊆ ℝ*
2		xpss12 5671	. 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 3900   × cxp 5654  ℝcr 11229  ℝ^*cxr 11372

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-un 4154  df-opab 5172  df-xp 5662  df-xr 11377

This theorem is used by:  ltrelxr  11400  xrsdsre  25163  ovolfioo  25821  ovolficc  25822  ovolficcss  25823  ovollb  25833  ovolicc2  25876  ovolfs2  25925  uniiccdif  25932  uniioovol  25933  uniiccvol  25934  uniioombllem2  25937  uniioombllem3a  25938  uniioombllem3  25939  uniioombllem4  25940  uniioombllem5  25941  uniioombl  25943  dyadmbllem  25953  opnmbllem  25955  icoreresf  38480  icoreelrn  38489  relowlpssretop  38492  opnmbllem0  38779  mblfinlem1  38780  mblfinlem2  38781  voliooicof  47221  ovolval3  47872  ovolval4lem2  47875  ovolval5lem2  47878  ovolval5lem3  47879  ovnovollem1  47881  ovnovollem2  47882