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Metamath Proof Explorer
Description: 'Less than' implies not equal. (Contributed by Glauco Siliprandi, 21-Nov-2020)
|
|
Ref |
Expression |
|
Hypotheses |
xrltned.1 |
|- ( ph -> A e. RR* ) |
|
|
xrltned.2 |
|- ( ph -> B e. RR* ) |
|
|
xrltned.3 |
|- ( ph -> A < B ) |
|
Assertion |
xrltned |
|- ( ph -> A =/= B ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
xrltned.1 |
|- ( ph -> A e. RR* ) |
| 2 |
|
xrltned.2 |
|- ( ph -> B e. RR* ) |
| 3 |
|
xrltned.3 |
|- ( ph -> A < B ) |
| 4 |
1 2 3
|
xrgtned |
|- ( ph -> B =/= A ) |
| 5 |
4
|
necomd |
|- ( ph -> A =/= B ) |