| Step |
Hyp |
Ref |
Expression |
| 1 |
|
n0 |
|- ( B =/= (/) <-> E. y y e. B ) |
| 2 |
|
ssltex1 |
|- ( A < A e. _V ) |
| 3 |
2
|
3ad2ant2 |
|- ( ( y e. B /\ A < A e. _V ) |
| 4 |
|
ssltex2 |
|- ( B < C e. _V ) |
| 5 |
4
|
3ad2ant3 |
|- ( ( y e. B /\ A < C e. _V ) |
| 6 |
|
ssltss1 |
|- ( A < A C_ No ) |
| 7 |
6
|
3ad2ant2 |
|- ( ( y e. B /\ A < A C_ No ) |
| 8 |
|
ssltss2 |
|- ( B < C C_ No ) |
| 9 |
8
|
3ad2ant3 |
|- ( ( y e. B /\ A < C C_ No ) |
| 10 |
7
|
3ad2ant1 |
|- ( ( ( y e. B /\ A < A C_ No ) |
| 11 |
|
simp2 |
|- ( ( ( y e. B /\ A < x e. A ) |
| 12 |
10 11
|
sseldd |
|- ( ( ( y e. B /\ A < x e. No ) |
| 13 |
|
ssltss2 |
|- ( A < B C_ No ) |
| 14 |
13
|
3ad2ant2 |
|- ( ( y e. B /\ A < B C_ No ) |
| 15 |
14
|
3ad2ant1 |
|- ( ( ( y e. B /\ A < B C_ No ) |
| 16 |
|
simp11 |
|- ( ( ( y e. B /\ A < y e. B ) |
| 17 |
15 16
|
sseldd |
|- ( ( ( y e. B /\ A < y e. No ) |
| 18 |
9
|
3ad2ant1 |
|- ( ( ( y e. B /\ A < C C_ No ) |
| 19 |
|
simp3 |
|- ( ( ( y e. B /\ A < z e. C ) |
| 20 |
18 19
|
sseldd |
|- ( ( ( y e. B /\ A < z e. No ) |
| 21 |
|
simp12 |
|- ( ( ( y e. B /\ A < A < |
| 22 |
21 11 16
|
ssltsepcd |
|- ( ( ( y e. B /\ A < x |
| 23 |
|
simp13 |
|- ( ( ( y e. B /\ A < B < |
| 24 |
23 16 19
|
ssltsepcd |
|- ( ( ( y e. B /\ A < y |
| 25 |
12 17 20 22 24
|
slttrd |
|- ( ( ( y e. B /\ A < x |
| 26 |
3 5 7 9 25
|
ssltd |
|- ( ( y e. B /\ A < A < |
| 27 |
26
|
3exp |
|- ( y e. B -> ( A < ( B < A < |
| 28 |
27
|
exlimiv |
|- ( E. y y e. B -> ( A < ( B < A < |
| 29 |
1 28
|
sylbi |
|- ( B =/= (/) -> ( A < ( B < A < |
| 30 |
29
|
3imp231 |
|- ( ( A < A < |