Description: Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri . Note: The more general version for extended reals is axlttri . Normally new proofs would use xrlttri . (New usage is discouraged.) (Contributed by NM, 13-Oct-2005)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax-pre-lttri | |- ( ( A e. RR /\ B e. RR ) -> ( A |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | cA | |- A |
|
| 1 | cr | |- RR |
|
| 2 | 0 1 | wcel | |- A e. RR |
| 3 | cB | |- B |
|
| 4 | 3 1 | wcel | |- B e. RR |
| 5 | 2 4 | wa | |- ( A e. RR /\ B e. RR ) |
| 6 | cltrr | |- |
|
| 7 | 0 3 6 | wbr | |- A |
| 8 | 0 3 | wceq | |- A = B |
| 9 | 3 0 6 | wbr | |- B |
| 10 | 8 9 | wo | |- ( A = B \/ B |
| 11 | 10 | wn | |- -. ( A = B \/ B |
| 12 | 7 11 | wb | |- ( A |
| 13 | 5 12 | wi | |- ( ( A e. RR /\ B e. RR ) -> ( A |