Step |
Hyp |
Ref |
Expression |
1 |
|
ltord.1 |
|- ( x = y -> A = B ) |
2 |
|
ltord.2 |
|- ( x = C -> A = M ) |
3 |
|
ltord.3 |
|- ( x = D -> A = N ) |
4 |
|
ltord.4 |
|- S C_ RR |
5 |
|
ltord.5 |
|- ( ( ph /\ x e. S ) -> A e. RR ) |
6 |
|
ltord.6 |
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> A < B ) ) |
7 |
1 2 3 4 5 6
|
ltordlem |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C < D -> M < N ) ) |
8 |
|
eqeq1 |
|- ( x = C -> ( x = D <-> C = D ) ) |
9 |
2
|
eqeq1d |
|- ( x = C -> ( A = N <-> M = N ) ) |
10 |
8 9
|
imbi12d |
|- ( x = C -> ( ( x = D -> A = N ) <-> ( C = D -> M = N ) ) ) |
11 |
10 3
|
vtoclg |
|- ( C e. S -> ( C = D -> M = N ) ) |
12 |
11
|
ad2antrl |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D -> M = N ) ) |
13 |
1 3 2 4 5 6
|
ltordlem |
|- ( ( ph /\ ( D e. S /\ C e. S ) ) -> ( D < C -> N < M ) ) |
14 |
13
|
ancom2s |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( D < C -> N < M ) ) |
15 |
12 14
|
orim12d |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( ( C = D \/ D < C ) -> ( M = N \/ N < M ) ) ) |
16 |
15
|
con3d |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( -. ( M = N \/ N < M ) -> -. ( C = D \/ D < C ) ) ) |
17 |
5
|
ralrimiva |
|- ( ph -> A. x e. S A e. RR ) |
18 |
2
|
eleq1d |
|- ( x = C -> ( A e. RR <-> M e. RR ) ) |
19 |
18
|
rspccva |
|- ( ( A. x e. S A e. RR /\ C e. S ) -> M e. RR ) |
20 |
17 19
|
sylan |
|- ( ( ph /\ C e. S ) -> M e. RR ) |
21 |
3
|
eleq1d |
|- ( x = D -> ( A e. RR <-> N e. RR ) ) |
22 |
21
|
rspccva |
|- ( ( A. x e. S A e. RR /\ D e. S ) -> N e. RR ) |
23 |
17 22
|
sylan |
|- ( ( ph /\ D e. S ) -> N e. RR ) |
24 |
20 23
|
anim12dan |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( M e. RR /\ N e. RR ) ) |
25 |
|
axlttri |
|- ( ( M e. RR /\ N e. RR ) -> ( M < N <-> -. ( M = N \/ N < M ) ) ) |
26 |
24 25
|
syl |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( M < N <-> -. ( M = N \/ N < M ) ) ) |
27 |
4
|
sseli |
|- ( C e. S -> C e. RR ) |
28 |
4
|
sseli |
|- ( D e. S -> D e. RR ) |
29 |
|
axlttri |
|- ( ( C e. RR /\ D e. RR ) -> ( C < D <-> -. ( C = D \/ D < C ) ) ) |
30 |
27 28 29
|
syl2an |
|- ( ( C e. S /\ D e. S ) -> ( C < D <-> -. ( C = D \/ D < C ) ) ) |
31 |
30
|
adantl |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C < D <-> -. ( C = D \/ D < C ) ) ) |
32 |
16 26 31
|
3imtr4d |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( M < N -> C < D ) ) |
33 |
7 32
|
impbid |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C < D <-> M < N ) ) |