Metamath Proof Explorer


Theorem nosepssdm

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 ANoBNoABxOn|AxBxdomA

Proof

Step Hyp Ref Expression
1 nosepne ANoBNoABAxOn|AxBxBxOn|AxBx
2 1 neneqd ANoBNoAB¬AxOn|AxBx=BxOn|AxBx
3 nodmord ANoOrddomA
4 3 3ad2ant1 ANoBNoABOrddomA
5 ordn2lp OrddomA¬domAxOn|AxBxxOn|AxBxdomA
6 4 5 syl ANoBNoAB¬domAxOn|AxBxxOn|AxBxdomA
7 imnan domAxOn|AxBx¬xOn|AxBxdomA¬domAxOn|AxBxxOn|AxBxdomA
8 6 7 sylibr ANoBNoABdomAxOn|AxBx¬xOn|AxBxdomA
9 8 imp ANoBNoABdomAxOn|AxBx¬xOn|AxBxdomA
10 ndmfv ¬xOn|AxBxdomAAxOn|AxBx=
11 9 10 syl ANoBNoABdomAxOn|AxBxAxOn|AxBx=
12 nosepeq ANoBNoABdomAxOn|AxBxAdomA=BdomA
13 simpl1 ANoBNoABdomAxOn|AxBxANo
14 ordirr OrddomA¬domAdomA
15 ndmfv ¬domAdomAAdomA=
16 13 3 14 15 4syl ANoBNoABdomAxOn|AxBxAdomA=
17 16 eqeq1d ANoBNoABdomAxOn|AxBxAdomA=BdomA=BdomA
18 eqcom =BdomABdomA=
19 17 18 bitrdi ANoBNoABdomAxOn|AxBxAdomA=BdomABdomA=
20 simpl2 ANoBNoABdomAxOn|AxBxBNo
21 nofun BNoFunB
22 20 21 syl ANoBNoABdomAxOn|AxBxFunB
23 nosgnn0 ¬1𝑜2𝑜
24 norn BNoranB1𝑜2𝑜
25 20 24 syl ANoBNoABdomAxOn|AxBxranB1𝑜2𝑜
26 25 sseld ANoBNoABdomAxOn|AxBxranB1𝑜2𝑜
27 23 26 mtoi ANoBNoABdomAxOn|AxBx¬ranB
28 funeldmb FunB¬ranBdomAdomBBdomA
29 22 27 28 syl2anc ANoBNoABdomAxOn|AxBxdomAdomBBdomA
30 29 necon2bbid ANoBNoABdomAxOn|AxBxBdomA=¬domAdomB
31 nodmord BNoOrddomB
32 31 3ad2ant2 ANoBNoABOrddomB
33 ordtr1 OrddomBdomAxOn|AxBxxOn|AxBxdomBdomAdomB
34 32 33 syl ANoBNoABdomAxOn|AxBxxOn|AxBxdomBdomAdomB
35 34 expdimp ANoBNoABdomAxOn|AxBxxOn|AxBxdomBdomAdomB
36 35 con3d ANoBNoABdomAxOn|AxBx¬domAdomB¬xOn|AxBxdomB
37 30 36 sylbid ANoBNoABdomAxOn|AxBxBdomA=¬xOn|AxBxdomB
38 19 37 sylbid ANoBNoABdomAxOn|AxBxAdomA=BdomA¬xOn|AxBxdomB
39 12 38 mpd ANoBNoABdomAxOn|AxBx¬xOn|AxBxdomB
40 ndmfv ¬xOn|AxBxdomBBxOn|AxBx=
41 39 40 syl ANoBNoABdomAxOn|AxBxBxOn|AxBx=
42 11 41 eqtr4d ANoBNoABdomAxOn|AxBxAxOn|AxBx=BxOn|AxBx
43 2 42 mtand ANoBNoAB¬domAxOn|AxBx
44 nosepon ANoBNoABxOn|AxBxOn
45 nodmon ANodomAOn
46 45 3ad2ant1 ANoBNoABdomAOn
47 ontri1 xOn|AxBxOndomAOnxOn|AxBxdomA¬domAxOn|AxBx
48 44 46 47 syl2anc ANoBNoABxOn|AxBxdomA¬domAxOn|AxBx
49 43 48 mpbird ANoBNoABxOn|AxBxdomA