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Metamath Proof Explorer

Theorem elfzlmr

Description: A member of a finite interval of integers is either its lower bound or its upper bound or an element of its interior. (Contributed by AV, 5-Feb-2021)

Ref Expression
Assertion elfzlmr 
|- ( K e. ( M ... N ) -> ( K = M \/ K e. ( ( M + 1 ) ..^ N ) \/ K = N ) )

Proof

Step Hyp Ref Expression
1 elfzuz2 
 |-  ( K e. ( M ... N ) -> N e. ( ZZ>= ` M ) )
2 fzpred 
 |-  ( N e. ( ZZ>= ` M ) -> ( M ... N ) = ( { M } u. ( ( M + 1 ) ... N ) ) )
3 2 eleq2d 
 |-  ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... N ) <-> K e. ( { M } u. ( ( M + 1 ) ... N ) ) ) )
4 elsni 
 |-  ( K e. { M } -> K = M )
5 elfzr 
 |-  ( K e. ( ( M + 1 ) ... N ) -> ( K e. ( ( M + 1 ) ..^ N ) \/ K = N ) )
6 4 5 orim12i 
 |-  ( ( K e. { M } \/ K e. ( ( M + 1 ) ... N ) ) -> ( K = M \/ ( K e. ( ( M + 1 ) ..^ N ) \/ K = N ) ) )
7 elun 
 |-  ( K e. ( { M } u. ( ( M + 1 ) ... N ) ) <-> ( K e. { M } \/ K e. ( ( M + 1 ) ... N ) ) )
8 3orass 
 |-  ( ( K = M \/ K e. ( ( M + 1 ) ..^ N ) \/ K = N ) <-> ( K = M \/ ( K e. ( ( M + 1 ) ..^ N ) \/ K = N ) ) )
9 6 7 8 3imtr4i 
 |-  ( K e. ( { M } u. ( ( M + 1 ) ... N ) ) -> ( K = M \/ K e. ( ( M + 1 ) ..^ N ) \/ K = N ) )
10 3 9 biimtrdi 
 |-  ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... N ) -> ( K = M \/ K e. ( ( M + 1 ) ..^ N ) \/ K = N ) ) )
11 1 10 mpcom 
 |-  ( K e. ( M ... N ) -> ( K = M \/ K e. ( ( M + 1 ) ..^ N ) \/ K = N ) )