Description: The slot for the orthocomplementation is not the slot for the base set in an extensible structure. Formerly part of proof for thlbas . (Contributed by AV, 11-Nov-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | basendxnocndx | |- ( Base ` ndx ) =/= ( oc ` ndx ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 1re | |- 1 e. RR |
|
| 2 | 1nn | |- 1 e. NN |
|
| 3 | 1nn0 | |- 1 e. NN0 |
|
| 4 | 1lt10 | |- 1 < ; 1 0 |
|
| 5 | 2 3 3 4 | declti | |- 1 < ; 1 1 |
| 6 | 1 5 | ltneii | |- 1 =/= ; 1 1 |
| 7 | basendx | |- ( Base ` ndx ) = 1 |
|
| 8 | ocndx | |- ( oc ` ndx ) = ; 1 1 |
|
| 9 | 7 8 | neeq12i | |- ( ( Base ` ndx ) =/= ( oc ` ndx ) <-> 1 =/= ; 1 1 ) |
| 10 | 6 9 | mpbir | |- ( Base ` ndx ) =/= ( oc ` ndx ) |