Metamath Proof Explorer

Theorem zltp1ne

Description: Integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023)

Ref Expression
Assertion zltp1ne 
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A + 1 ) < B <-> ( A < B /\ B =/= ( A + 1 ) ) ) )

Proof

Step Hyp Ref Expression
1 zre 
 |-  ( A e. ZZ -> A e. RR )
2 zre 
 |-  ( B e. ZZ -> B e. RR )
3 peano2re 
 |-  ( A e. RR -> ( A + 1 ) e. RR )
4 ltlen 
 |-  ( ( ( A + 1 ) e. RR /\ B e. RR ) -> ( ( A + 1 ) < B <-> ( ( A + 1 ) <_ B /\ B =/= ( A + 1 ) ) ) )
5 3 4 sylan 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( A + 1 ) < B <-> ( ( A + 1 ) <_ B /\ B =/= ( A + 1 ) ) ) )
6 1 2 5 syl2an 
 |-  ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A + 1 ) < B <-> ( ( A + 1 ) <_ B /\ B =/= ( A + 1 ) ) ) )
7 zltp1le 
 |-  ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B <-> ( A + 1 ) <_ B ) )
8 7 anbi1d 
 |-  ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A < B /\ B =/= ( A + 1 ) ) <-> ( ( A + 1 ) <_ B /\ B =/= ( A + 1 ) ) ) )
9 6 8 bitr4d 
 |-  ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A + 1 ) < B <-> ( A < B /\ B =/= ( A + 1 ) ) ) )