Description: The slot for the orthocomplementation is not the slot for the order in an extensible structure. Formerly part of proof for thlle . (Contributed by AV, 11-Nov-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | plendxnocndx | |- ( le ` ndx ) =/= ( oc ` ndx ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 10re | |- ; 1 0 e. RR |
|
| 2 | 1nn0 | |- 1 e. NN0 |
|
| 3 | 0nn0 | |- 0 e. NN0 |
|
| 4 | 1nn | |- 1 e. NN |
|
| 5 | 0lt1 | |- 0 < 1 |
|
| 6 | 2 3 4 5 | declt | |- ; 1 0 < ; 1 1 |
| 7 | 1 6 | ltneii | |- ; 1 0 =/= ; 1 1 |
| 8 | plendx | |- ( le ` ndx ) = ; 1 0 |
|
| 9 | ocndx | |- ( oc ` ndx ) = ; 1 1 |
|
| 10 | 8 9 | neeq12i | |- ( ( le ` ndx ) =/= ( oc ` ndx ) <-> ; 1 0 =/= ; 1 1 ) |
| 11 | 7 10 | mpbir | |- ( le ` ndx ) =/= ( oc ` ndx ) |