Metamath Proof Explorer

Theorem scandxnbasendx

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

Ref Expression
Assertion scandxnbasendx 
|- ( Scalar ` ndx ) =/= ( Base ` ndx )

Proof

Step Hyp Ref Expression
1 1re 
 |-  1 e. RR
2 1lt5 
 |-  1 < 5
3 1 2 gtneii 
 |-  5 =/= 1
4 scandx 
 |-  ( Scalar ` ndx ) = 5
5 basendx 
 |-  ( Base ` ndx ) = 1
6 4 5 neeq12i 
 |-  ( ( Scalar ` ndx ) =/= ( Base ` ndx ) <-> 5 =/= 1 )
7 3 6 mpbir 
 |-  ( Scalar ` ndx ) =/= ( Base ` ndx )