theorem ltne (a b: nat): $ a < b -> a != b $;
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltirr | ~b < b |
|
2 | lteq1 | a = b -> (a < b <-> b < b) |
|
3 | 2 | bi1d | a = b -> a < b -> b < b |
4 | 3 | com12 | a < b -> a = b -> b < b |
5 | 4 | con3d | a < b -> ~b < b -> ~a = b |
6 | 5 | conv ne | a < b -> ~b < b -> a != b |
7 | 1, 6 | mpi | a < b -> a != b |