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SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
Mathbox for Mario Carneiro
Derangements and the Subfactorial
derang0
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derangsn
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Theorem
derang0
Description:
The derangement number of the empty set.
(Contributed by
Mario Carneiro
, 19-Jan-2015)
Ref
Expression
Hypothesis
derang.d
⊢
D
=
x
∈
Fin
⟼
f
|
f
:
x
⟶
1-1 onto
x
∧
∀
y
∈
x
f
⁡
y
≠
y
Assertion
derang0
⊢
D
⁡
∅
=
1
Proof
Step
Hyp
Ref
Expression
1
derang.d
⊢
D
=
x
∈
Fin
⟼
f
|
f
:
x
⟶
1-1 onto
x
∧
∀
y
∈
x
f
⁡
y
≠
y
2
0fin
⊢
∅
∈
Fin
3
1
derangval
⊢
∅
∈
Fin
→
D
⁡
∅
=
f
|
f
:
∅
⟶
1-1 onto
∅
∧
∀
y
∈
∅
f
⁡
y
≠
y
4
2
3
ax-mp
⊢
D
⁡
∅
=
f
|
f
:
∅
⟶
1-1 onto
∅
∧
∀
y
∈
∅
f
⁡
y
≠
y
5
ral0
⊢
∀
y
∈
∅
f
⁡
y
≠
y
6
5
biantru
⊢
f
:
∅
⟶
1-1 onto
∅
↔
f
:
∅
⟶
1-1 onto
∅
∧
∀
y
∈
∅
f
⁡
y
≠
y
7
eqid
⊢
∅
=
∅
8
f1o00
⊢
f
:
∅
⟶
1-1 onto
∅
↔
f
=
∅
∧
∅
=
∅
9
7
8
mpbiran2
⊢
f
:
∅
⟶
1-1 onto
∅
↔
f
=
∅
10
6
9
bitr3i
⊢
f
:
∅
⟶
1-1 onto
∅
∧
∀
y
∈
∅
f
⁡
y
≠
y
↔
f
=
∅
11
10
abbii
⊢
f
|
f
:
∅
⟶
1-1 onto
∅
∧
∀
y
∈
∅
f
⁡
y
≠
y
=
f
|
f
=
∅
12
df-sn
⊢
∅
=
f
|
f
=
∅
13
11
12
eqtr4i
⊢
f
|
f
:
∅
⟶
1-1 onto
∅
∧
∀
y
∈
∅
f
⁡
y
≠
y
=
∅
14
13
fveq2i
⊢
f
|
f
:
∅
⟶
1-1 onto
∅
∧
∀
y
∈
∅
f
⁡
y
≠
y
=
∅
15
0ex
⊢
∅
∈
V
16
hashsng
⊢
∅
∈
V
→
∅
=
1
17
15
16
ax-mp
⊢
∅
=
1
18
4
14
17
3eqtri
⊢
D
⁡
∅
=
1