Theorem axresscn

Description: The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-resscn . (Contributed by NM, 1-Mar-1995) (Proof shortened by Andrew Salmon, 12-Aug-2011) (New usage is discouraged.)

	Ref	Expression
Assertion	axresscn	$$⊢ℝ\subseteq ℂ$$

Proof

Step	Hyp	Ref	Expression
1		0r	$$⊢0_{𝑹}\in 𝑹$$
2		snssi	$$⊢0_{𝑹}\in 𝑹\to \left\{0_{𝑹}\right\}\subseteq 𝑹$$
3		xpss2	$$⊢\left\{0_{𝑹}\right\}\subseteq 𝑹\to 𝑹\times \left\{0_{𝑹}\right\}\subseteq 𝑹\times 𝑹$$
4	1 2 3	mp2b	$$⊢𝑹\times \left\{0_{𝑹}\right\}\subseteq 𝑹\times 𝑹$$
5		df-r	$$⊢ℝ=𝑹\times \left\{0_{𝑹}\right\}$$
6		df-c	$$⊢ℂ=𝑹\times 𝑹$$
7	4 5 6	3sstr4i	$$⊢ℝ\subseteq ℂ$$