Metamath Proof Explorer

Theorem lem1

Description: A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015)

Ref Expression
Assertion lem1 
|- ( A e. RR -> ( A - 1 ) <_ A )

Proof

Step Hyp Ref Expression
1 ltm1 
 |-  ( A e. RR -> ( A - 1 ) < A )
2 peano2rem 
 |-  ( A e. RR -> ( A - 1 ) e. RR )
3 ltle 
 |-  ( ( ( A - 1 ) e. RR /\ A e. RR ) -> ( ( A - 1 ) < A -> ( A - 1 ) <_ A ) )
4 2 3 mpancom 
 |-  ( A e. RR -> ( ( A - 1 ) < A -> ( A - 1 ) <_ A ) )
5 1 4 mpd 
 |-  ( A e. RR -> ( A - 1 ) <_ A )