Database
ZF (ZERMELO-FRAENKEL) SET THEORY
ZF Set Theory - add the Axiom of Power Sets
The Predecessor Class
predin
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predun
Metamath Proof Explorer
Ascii
Unicode
Theorem
predin
Description:
Intersection law for predecessor classes.
(Contributed by
Scott Fenton
, 29-Mar-2011)
Ref
Expression
Assertion
predin
⊢
Pred
R
A
∩
B
X
=
Pred
R
A
X
∩
Pred
R
B
X
Proof
Step
Hyp
Ref
Expression
1
inindir
⊢
A
∩
B
∩
R
-1
X
=
A
∩
R
-1
X
∩
B
∩
R
-1
X
2
df-pred
⊢
Pred
R
A
∩
B
X
=
A
∩
B
∩
R
-1
X
3
df-pred
⊢
Pred
R
A
X
=
A
∩
R
-1
X
4
df-pred
⊢
Pred
R
B
X
=
B
∩
R
-1
X
5
3
4
ineq12i
⊢
Pred
R
A
X
∩
Pred
R
B
X
=
A
∩
R
-1
X
∩
B
∩
R
-1
X
6
1
2
5
3eqtr4i
⊢
Pred
R
A
∩
B
X
=
Pred
R
A
X
∩
Pred
R
B
X