Theorem rpltrp

Description: For all positive real numbers there is a smaller positive real number. (Contributed by AV, 5-Sep-2020)

	Ref	Expression
Assertion	rpltrp	|- A. x e. RR+ E. y e. RR+ y < x

Proof

Step	Hyp	Ref	Expression
1		rphalfcl	|- ( x e. RR+ -> ( x / 2 ) e. RR+ )
2		breq1	|- ( y = ( x / 2 ) -> ( y < x <-> ( x / 2 ) < x ) )
3	2	adantl	|- ( ( x e. RR+ /\ y = ( x / 2 ) ) -> ( y < x <-> ( x / 2 ) < x ) )
4		rphalflt	|- ( x e. RR+ -> ( x / 2 ) < x )
5	1 3 4	rspcedvd	|- ( x e. RR+ -> E. y e. RR+ y < x )
6	5	rgen	|- A. x e. RR+ E. y e. RR+ y < x