Metamath Proof Explorer

Theorem tsetndxnbasendx

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

Ref Expression
Assertion tsetndxnbasendx 
|- ( TopSet ` ndx ) =/= ( Base ` ndx )

Proof

Step Hyp Ref Expression
1 1re 
 |-  1 e. RR
2 1lt9 
 |-  1 < 9
3 1 2 gtneii 
 |-  9 =/= 1
4 tsetndx 
 |-  ( TopSet ` ndx ) = 9
5 basendx 
 |-  ( Base ` ndx ) = 1
6 4 5 neeq12i 
 |-  ( ( TopSet ` ndx ) =/= ( Base ` ndx ) <-> 9 =/= 1 )
7 3 6 mpbir 
 |-  ( TopSet ` ndx ) =/= ( Base ` ndx )