Metamath Proof Explorer

Theorem dsndxnbasendx

Description: The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024)

Ref Expression
Assertion dsndxnbasendx 
|- ( dist ` ndx ) =/= ( Base ` ndx )

Proof

Step Hyp Ref Expression
1 1re 
 |-  1 e. RR
2 1nn 
 |-  1 e. NN
3 2nn0 
 |-  2 e. NN0
4 1nn0 
 |-  1 e. NN0
5 1lt10 
 |-  1 < ; 1 0
6 2 3 4 5 declti 
 |-  1 < ; 1 2
7 1 6 gtneii 
 |-  ; 1 2 =/= 1
8 dsndx 
 |-  ( dist ` ndx ) = ; 1 2
9 basendx 
 |-  ( Base ` ndx ) = 1
10 8 9 neeq12i 
 |-  ( ( dist ` ndx ) =/= ( Base ` ndx ) <-> ; 1 2 =/= 1 )
11 7 10 mpbir 
 |-  ( dist ` ndx ) =/= ( Base ` ndx )