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Metamath Proof Explorer

Theorem elfz1eq

Description: Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005)

Ref Expression
Assertion elfz1eq 
|- ( K e. ( N ... N ) -> K = N )

Proof

Step Hyp Ref Expression
1 elfzle2 
 |-  ( K e. ( N ... N ) -> K <_ N )
2 elfzle1 
 |-  ( K e. ( N ... N ) -> N <_ K )
3 elfzelz 
 |-  ( K e. ( N ... N ) -> K e. ZZ )
4 elfzel2 
 |-  ( K e. ( N ... N ) -> N e. ZZ )
5 zre 
 |-  ( K e. ZZ -> K e. RR )
6 zre 
 |-  ( N e. ZZ -> N e. RR )
7 letri3 
 |-  ( ( K e. RR /\ N e. RR ) -> ( K = N <-> ( K <_ N /\ N <_ K ) ) )
8 5 6 7 syl2an 
 |-  ( ( K e. ZZ /\ N e. ZZ ) -> ( K = N <-> ( K <_ N /\ N <_ K ) ) )
9 3 4 8 syl2anc 
 |-  ( K e. ( N ... N ) -> ( K = N <-> ( K <_ N /\ N <_ K ) ) )
10 1 2 9 mpbir2and 
 |-  ( K e. ( N ... N ) -> K = N )