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Metamath Proof Explorer

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