Database
SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
Mathbox for Scott Fenton
Surreal numbers: Conway cuts
scutcut
Next ⟩
scutcl
Metamath Proof Explorer
Ascii
Unicode
Theorem
scutcut
Description:
Cut properties of the surreal cut operation.
(Contributed by
Scott Fenton
, 8-Dec-2021)
Ref
Expression
Assertion
scutcut
⊢
A
≪
s
B
→
A
|
s
B
∈
No
∧
A
≪
s
A
|
s
B
∧
A
|
s
B
≪
s
B
Proof
Step
Hyp
Ref
Expression
1
scutval
⊢
A
≪
s
B
→
A
|
s
B
=
ι
x
∈
y
∈
No
|
A
≪
s
y
∧
y
≪
s
B
|
bday
⁡
x
=
⋂
bday
y
∈
No
|
A
≪
s
y
∧
y
≪
s
B
2
conway
⊢
A
≪
s
B
→
∃!
x
∈
y
∈
No
|
A
≪
s
y
∧
y
≪
s
B
bday
⁡
x
=
⋂
bday
y
∈
No
|
A
≪
s
y
∧
y
≪
s
B
3
riotacl
⊢
∃!
x
∈
y
∈
No
|
A
≪
s
y
∧
y
≪
s
B
bday
⁡
x
=
⋂
bday
y
∈
No
|
A
≪
s
y
∧
y
≪
s
B
→
ι
x
∈
y
∈
No
|
A
≪
s
y
∧
y
≪
s
B
|
bday
⁡
x
=
⋂
bday
y
∈
No
|
A
≪
s
y
∧
y
≪
s
B
∈
y
∈
No
|
A
≪
s
y
∧
y
≪
s
B
4
2
3
syl
⊢
A
≪
s
B
→
ι
x
∈
y
∈
No
|
A
≪
s
y
∧
y
≪
s
B
|
bday
⁡
x
=
⋂
bday
y
∈
No
|
A
≪
s
y
∧
y
≪
s
B
∈
y
∈
No
|
A
≪
s
y
∧
y
≪
s
B
5
1
4
eqeltrd
⊢
A
≪
s
B
→
A
|
s
B
∈
y
∈
No
|
A
≪
s
y
∧
y
≪
s
B
6
sneq
⊢
y
=
A
|
s
B
→
y
=
A
|
s
B
7
6
breq2d
⊢
y
=
A
|
s
B
→
A
≪
s
y
↔
A
≪
s
A
|
s
B
8
6
breq1d
⊢
y
=
A
|
s
B
→
y
≪
s
B
↔
A
|
s
B
≪
s
B
9
7
8
anbi12d
⊢
y
=
A
|
s
B
→
A
≪
s
y
∧
y
≪
s
B
↔
A
≪
s
A
|
s
B
∧
A
|
s
B
≪
s
B
10
9
elrab
⊢
A
|
s
B
∈
y
∈
No
|
A
≪
s
y
∧
y
≪
s
B
↔
A
|
s
B
∈
No
∧
A
≪
s
A
|
s
B
∧
A
|
s
B
≪
s
B
11
3anass
⊢
A
|
s
B
∈
No
∧
A
≪
s
A
|
s
B
∧
A
|
s
B
≪
s
B
↔
A
|
s
B
∈
No
∧
A
≪
s
A
|
s
B
∧
A
|
s
B
≪
s
B
12
10
11
bitr4i
⊢
A
|
s
B
∈
y
∈
No
|
A
≪
s
y
∧
y
≪
s
B
↔
A
|
s
B
∈
No
∧
A
≪
s
A
|
s
B
∧
A
|
s
B
≪
s
B
13
5
12
sylib
⊢
A
≪
s
B
→
A
|
s
B
∈
No
∧
A
≪
s
A
|
s
B
∧
A
|
s
B
≪
s
B