Description: Given two non-equal surreals, their separator is less-than or equal to the domain of one of them. Part of Lemma 2.1.1 of Lipparini p. 3. (Contributed by Scott Fenton, 6-Dec-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | nosepssdm | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nosepne | |
|
2 | 1 | neneqd | |
3 | nodmord | |
|
4 | 3 | 3ad2ant1 | |
5 | ordn2lp | |
|
6 | 4 5 | syl | |
7 | imnan | |
|
8 | 6 7 | sylibr | |
9 | 8 | imp | |
10 | ndmfv | |
|
11 | 9 10 | syl | |
12 | nosepeq | |
|
13 | simpl1 | |
|
14 | ordirr | |
|
15 | ndmfv | |
|
16 | 13 3 14 15 | 4syl | |
17 | 16 | eqeq1d | |
18 | eqcom | |
|
19 | 17 18 | bitrdi | |
20 | simpl2 | |
|
21 | nofun | |
|
22 | 20 21 | syl | |
23 | nosgnn0 | |
|
24 | norn | |
|
25 | 20 24 | syl | |
26 | 25 | sseld | |
27 | 23 26 | mtoi | |
28 | funeldmb | |
|
29 | 22 27 28 | syl2anc | |
30 | 29 | necon2bbid | |
31 | nodmord | |
|
32 | 31 | 3ad2ant2 | |
33 | ordtr1 | |
|
34 | 32 33 | syl | |
35 | 34 | expdimp | |
36 | 35 | con3d | |
37 | 30 36 | sylbid | |
38 | 19 37 | sylbid | |
39 | 12 38 | mpd | |
40 | ndmfv | |
|
41 | 39 40 | syl | |
42 | 11 41 | eqtr4d | |
43 | 2 42 | mtand | |
44 | nosepon | |
|
45 | nodmon | |
|
46 | 45 | 3ad2ant1 | |
47 | ontri1 | |
|
48 | 44 46 47 | syl2anc | |
49 | 43 48 | mpbird | |