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 A No B No A B x On | A x B x dom A

Proof

Step Hyp Ref Expression
1 nosepne A No B No A B A x On | A x B x B x On | A x B x
2 1 neneqd A No B No A B ¬ A x On | A x B x = B x On | A x B x
3 nodmord A No Ord dom A
4 3 3ad2ant1 A No B No A B Ord dom A
5 ordn2lp Ord dom A ¬ dom A x On | A x B x x On | A x B x dom A
6 4 5 syl A No B No A B ¬ dom A x On | A x B x x On | A x B x dom A
7 imnan dom A x On | A x B x ¬ x On | A x B x dom A ¬ dom A x On | A x B x x On | A x B x dom A
8 6 7 sylibr A No B No A B dom A x On | A x B x ¬ x On | A x B x dom A
9 8 imp A No B No A B dom A x On | A x B x ¬ x On | A x B x dom A
10 ndmfv ¬ x On | A x B x dom A A x On | A x B x =
11 9 10 syl A No B No A B dom A x On | A x B x A x On | A x B x =
12 nosepeq A No B No A B dom A x On | A x B x A dom A = B dom A
13 simpl1 A No B No A B dom A x On | A x B x A No
14 13 3 syl A No B No A B dom A x On | A x B x Ord dom A
15 ordirr Ord dom A ¬ dom A dom A
16 ndmfv ¬ dom A dom A A dom A =
17 14 15 16 3syl A No B No A B dom A x On | A x B x A dom A =
18 17 eqeq1d A No B No A B dom A x On | A x B x A dom A = B dom A = B dom A
19 eqcom = B dom A B dom A =
20 18 19 bitrdi A No B No A B dom A x On | A x B x A dom A = B dom A B dom A =
21 simpl2 A No B No A B dom A x On | A x B x B No
22 nofun B No Fun B
23 21 22 syl A No B No A B dom A x On | A x B x Fun B
24 nosgnn0 ¬ 1 𝑜 2 𝑜
25 norn B No ran B 1 𝑜 2 𝑜
26 21 25 syl A No B No A B dom A x On | A x B x ran B 1 𝑜 2 𝑜
27 26 sseld A No B No A B dom A x On | A x B x ran B 1 𝑜 2 𝑜
28 24 27 mtoi A No B No A B dom A x On | A x B x ¬ ran B
29 funeldmb Fun B ¬ ran B dom A dom B B dom A
30 23 28 29 syl2anc A No B No A B dom A x On | A x B x dom A dom B B dom A
31 30 necon2bbid A No B No A B dom A x On | A x B x B dom A = ¬ dom A dom B
32 nodmord B No Ord dom B
33 32 3ad2ant2 A No B No A B Ord dom B
34 ordtr1 Ord dom B dom A x On | A x B x x On | A x B x dom B dom A dom B
35 33 34 syl A No B No A B dom A x On | A x B x x On | A x B x dom B dom A dom B
36 35 expdimp A No B No A B dom A x On | A x B x x On | A x B x dom B dom A dom B
37 36 con3d A No B No A B dom A x On | A x B x ¬ dom A dom B ¬ x On | A x B x dom B
38 31 37 sylbid A No B No A B dom A x On | A x B x B dom A = ¬ x On | A x B x dom B
39 20 38 sylbid A No B No A B dom A x On | A x B x A dom A = B dom A ¬ x On | A x B x dom B
40 12 39 mpd A No B No A B dom A x On | A x B x ¬ x On | A x B x dom B
41 ndmfv ¬ x On | A x B x dom B B x On | A x B x =
42 40 41 syl A No B No A B dom A x On | A x B x B x On | A x B x =
43 11 42 eqtr4d A No B No A B dom A x On | A x B x A x On | A x B x = B x On | A x B x
44 2 43 mtand A No B No A B ¬ dom A x On | A x B x
45 nosepon A No B No A B x On | A x B x On
46 nodmon A No dom A On
47 46 3ad2ant1 A No B No A B dom A On
48 ontri1 x On | A x B x On dom A On x On | A x B x dom A ¬ dom A x On | A x B x
49 45 47 48 syl2anc A No B No A B x On | A x B x dom A ¬ dom A x On | A x B x
50 44 49 mpbird A No B No A B x On | A x B x dom A