Theorem rexpssxrxp 8020

Description: The Cartesian product of standard reals are a subset of the Cartesian product of extended reals (common case). (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 8019	. 2 ⊢ ℝ ⊆ ℝ*
2		xpss12 4748	. 2 ⊢ ((ℝ ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
3	1, 1, 2	mp2an 426	1 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)

Syntax hints:   ⊆ wss 3144   × cxp 4639  ℝcr 7828  ℝ^*cxr 8009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-opab 4080  df-xp 4647  df-xr 8014

This theorem is referenced by:  ltrelxr  8036