Description: The slot for the distance function is not the slot for the topology in an extensible structure. Formerly part of proof for tngds . (Contributed by AV, 29-Oct-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | dsndxntsetndx | |- ( dist ` ndx ) =/= ( TopSet ` ndx ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 9re | |- 9 e. RR |
|
| 2 | 1nn | |- 1 e. NN |
|
| 3 | 2nn0 | |- 2 e. NN0 |
|
| 4 | 9nn0 | |- 9 e. NN0 |
|
| 5 | 9lt10 | |- 9 < ; 1 0 |
|
| 6 | 2 3 4 5 | declti | |- 9 < ; 1 2 |
| 7 | 1 6 | gtneii | |- ; 1 2 =/= 9 |
| 8 | dsndx | |- ( dist ` ndx ) = ; 1 2 |
|
| 9 | tsetndx | |- ( TopSet ` ndx ) = 9 |
|
| 10 | 8 9 | neeq12i | |- ( ( dist ` ndx ) =/= ( TopSet ` ndx ) <-> ; 1 2 =/= 9 ) |
| 11 | 7 10 | mpbir | |- ( dist ` ndx ) =/= ( TopSet ` ndx ) |