Theorem rucALT

Description: Alternate proof of ruc . This proof is a simple corollary of rpnnen , which determines the exact cardinality of the reals. For an alternate proof discussed at mmcomplex.html#uncountable , see ruc . (Contributed by NM, 13-Oct-2004) (Revised by Mario Carneiro, 13-May-2013) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	rucALT	|- NN ~< RR

Proof

Step	Hyp	Ref	Expression
1		nnex	|- NN e. _V
2	1	canth2	|- NN ~< ~P NN
3		rpnnen	|- RR ~~ ~P NN
4	3	ensymi	|- ~P NN ~~ RR
5		sdomentr	|- ( ( NN ~< ~P NN /\ ~P NN ~~ RR ) -> NN ~< RR )
6	2 4 5	mp2an	|- NN ~< RR