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Metamath Proof Explorer

Theorem elfzp1

Description: Append an element to a finite set of sequential integers. (Contributed by NM, 19-Sep-2005) (Proof shortened by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion elfzp1 
|- ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... ( N + 1 ) ) <-> ( K e. ( M ... N ) \/ K = ( N + 1 ) ) ) )

Proof

Step Hyp Ref Expression
1 fzsuc 
 |-  ( N e. ( ZZ>= ` M ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )
2 1 eleq2d 
 |-  ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... ( N + 1 ) ) <-> K e. ( ( M ... N ) u. { ( N + 1 ) } ) ) )
3 elun 
 |-  ( K e. ( ( M ... N ) u. { ( N + 1 ) } ) <-> ( K e. ( M ... N ) \/ K e. { ( N + 1 ) } ) )
4 ovex 
 |-  ( N + 1 ) e. _V
5 4 elsn2 
 |-  ( K e. { ( N + 1 ) } <-> K = ( N + 1 ) )
6 5 orbi2i 
 |-  ( ( K e. ( M ... N ) \/ K e. { ( N + 1 ) } ) <-> ( K e. ( M ... N ) \/ K = ( N + 1 ) ) )
7 3 6 bitri 
 |-  ( K e. ( ( M ... N ) u. { ( N + 1 ) } ) <-> ( K e. ( M ... N ) \/ K = ( N + 1 ) ) )
8 2 7 bitrdi 
 |-  ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... ( N + 1 ) ) <-> ( K e. ( M ... N ) \/ K = ( N + 1 ) ) ) )