Metamath Proof Explorer

Theorem letrp1

Description: A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005)

Ref Expression
Assertion letrp1 
|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> A <_ ( B + 1 ) )

Proof

Step Hyp Ref Expression
1 ltp1 
 |-  ( B e. RR -> B < ( B + 1 ) )
2 1 adantl 
 |-  ( ( A e. RR /\ B e. RR ) -> B < ( B + 1 ) )
3 peano2re 
 |-  ( B e. RR -> ( B + 1 ) e. RR )
4 3 ancli 
 |-  ( B e. RR -> ( B e. RR /\ ( B + 1 ) e. RR ) )
5 lelttr 
 |-  ( ( A e. RR /\ B e. RR /\ ( B + 1 ) e. RR ) -> ( ( A <_ B /\ B < ( B + 1 ) ) -> A < ( B + 1 ) ) )
6 5 3expb 
 |-  ( ( A e. RR /\ ( B e. RR /\ ( B + 1 ) e. RR ) ) -> ( ( A <_ B /\ B < ( B + 1 ) ) -> A < ( B + 1 ) ) )
7 4 6 sylan2 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( A <_ B /\ B < ( B + 1 ) ) -> A < ( B + 1 ) ) )
8 2 7 mpan2d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A <_ B -> A < ( B + 1 ) ) )
9 8 3impia 
 |-  ( ( A e. RR /\ B e. RR /\ A <_ B ) -> A < ( B + 1 ) )
10 ltle 
 |-  ( ( A e. RR /\ ( B + 1 ) e. RR ) -> ( A < ( B + 1 ) -> A <_ ( B + 1 ) ) )
11 3 10 sylan2 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A < ( B + 1 ) -> A <_ ( B + 1 ) ) )
12 11 3adant3 
 |-  ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( A < ( B + 1 ) -> A <_ ( B + 1 ) ) )
13 9 12 mpd 
 |-  ( ( A e. RR /\ B e. RR /\ A <_ B ) -> A <_ ( B + 1 ) )