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 ⊆ ℂ

Proof

Step	Hyp	Ref	Expression
1		0r	⊢ 0R ∈ R
2		snssi	⊢ ( 0R ∈ R → { 0R } ⊆ R )
3		xpss2	⊢ ( { 0R } ⊆ R → ( R × { 0R } ) ⊆ ( R × R ) )
4	1 2 3	mp2b	⊢ ( R × { 0R } ) ⊆ ( R × R )
5		df-r	⊢ ℝ = ( R × { 0R } )
6		df-c	⊢ ℂ = ( R × R )
7	4 5 6	3sstr4i	⊢ ℝ ⊆ ℂ