Metamath Proof Explorer

Theorem p1le

Description: A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005)

Ref Expression
Assertion p1le 
|- ( ( A e. RR /\ B e. RR /\ ( A + 1 ) <_ B ) -> A <_ B )

Proof

Step Hyp Ref Expression
1 lep1 
 |-  ( A e. RR -> A <_ ( A + 1 ) )
2 1 adantr 
 |-  ( ( A e. RR /\ B e. RR ) -> A <_ ( A + 1 ) )
3 peano2re 
 |-  ( A e. RR -> ( A + 1 ) e. RR )
4 3 ancli 
 |-  ( A e. RR -> ( A e. RR /\ ( A + 1 ) e. RR ) )
5 letr 
 |-  ( ( A e. RR /\ ( A + 1 ) e. RR /\ B e. RR ) -> ( ( A <_ ( A + 1 ) /\ ( A + 1 ) <_ B ) -> A <_ B ) )
6 5 3expa 
 |-  ( ( ( A e. RR /\ ( A + 1 ) e. RR ) /\ B e. RR ) -> ( ( A <_ ( A + 1 ) /\ ( A + 1 ) <_ B ) -> A <_ B ) )
7 4 6 sylan 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( A <_ ( A + 1 ) /\ ( A + 1 ) <_ B ) -> A <_ B ) )
8 2 7 mpand 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( A + 1 ) <_ B -> A <_ B ) )
9 8 3impia 
 |-  ( ( A e. RR /\ B e. RR /\ ( A + 1 ) <_ B ) -> A <_ B )