Theorem rr2sscn2 46139

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 11172	. 2 ⊢ ℝ ⊆ ℂ
2		xpss12 5678	. 2 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
3	1, 1, 2	mp2an 705	1 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)

This proof depends on syntax axioms:   ⊆ wss 3906   × cxp 5661  ℂcc 11113  ℝcr 11114

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-resscn 11172

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ss 3923  df-opab 5176  df-xp 5669

This theorem is used by:  ovolval2lem  47415  ovolval2  47416  ovolval3  47419