Theorem reltre

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

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

Proof

Step	Hyp	Ref	Expression
1		peano2rem	|- ( x e. RR -> ( x - 1 ) e. RR )
2		breq1	|- ( y = ( x - 1 ) -> ( y < x <-> ( x - 1 ) < x ) )
3	2	adantl	|- ( ( x e. RR /\ y = ( x - 1 ) ) -> ( y < x <-> ( x - 1 ) < x ) )
4		ltm1	|- ( x e. RR -> ( x - 1 ) < 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