Theorem rexpssxrxp 8124

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 8123	. 2 ⊢ ℝ ⊆ ℝ*
2		xpss12 4786	. 2 ⊢ ((ℝ ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
3	1, 1, 2	mp2an 426	1 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)

Syntax hints:   ⊆ wss 3167   × cxp 4677  ℝcr 7931  ℝ^*cxr 8113

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-opab 4110  df-xp 4685  df-xr 8118

This theorem is referenced by:  ltrelxr  8140