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	|- RR C_ CC

Proof

Step	Hyp	Ref	Expression
1		0r	|- 0R e. R.
2		snssi	|- ( 0R e. R. -> { 0R } C_ R. )
3		xpss2	|- ( { 0R } C_ R. -> ( R. X. { 0R } ) C_ ( R. X. R. ) )
4	1 2 3	mp2b	|- ( R. X. { 0R } ) C_ ( R. X. R. )
5		df-r	|- RR = ( R. X. { 0R } )
6		df-c	|- CC = ( R. X. R. )
7	4 5 6	3sstr4i	|- RR C_ CC