Theorem rr2sscn2 41641

Description: The cartesian square of ℝ is a subset of the cartesian square of ℂ. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Assertion

Ref	Expression
rr2sscn2	⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)

Proof of Theorem rr2sscn2

Step	Hyp	Ref	Expression
1		ax-resscn 10596	. 2 ⊢ ℝ ⊆ ℂ
2		xpss12 5572	. 2 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
3	1, 1, 2	mp2an 690	1 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)

Syntax hints:   ⊆ wss 3938   × cxp 5555  ℂcc 10537  ℝcr 10538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-resscn 10596

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-in 3945  df-ss 3954  df-opab 5131  df-xp 5563

This theorem is referenced by:  ovolval2lem  42932  ovolval2  42933  ovolval3  42936