Step |
Hyp |
Ref |
Expression |
1 |
|
xrlttri |
|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -. ( A = B \/ B < A ) ) ) |
2 |
|
ioran |
|- ( -. ( A = B \/ B < A ) <-> ( -. A = B /\ -. B < A ) ) |
3 |
2
|
biancomi |
|- ( -. ( A = B \/ B < A ) <-> ( -. B < A /\ -. A = B ) ) |
4 |
1 3
|
bitrdi |
|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> ( -. B < A /\ -. A = B ) ) ) |
5 |
|
xrlenlt |
|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) ) |
6 |
|
nesym |
|- ( B =/= A <-> -. A = B ) |
7 |
6
|
a1i |
|- ( ( A e. RR* /\ B e. RR* ) -> ( B =/= A <-> -. A = B ) ) |
8 |
5 7
|
anbi12d |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A <_ B /\ B =/= A ) <-> ( -. B < A /\ -. A = B ) ) ) |
9 |
4 8
|
bitr4d |
|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> ( A <_ B /\ B =/= A ) ) ) |