Database
ZF (ZERMELO-FRAENKEL) SET THEORY
ZF Set Theory - start with the Axiom of Extensionality
Unordered and ordered pairs
preq1
Next ⟩
preq2
Metamath Proof Explorer
Ascii
Unicode
Theorem
preq1
Description:
Equality theorem for unordered pairs.
(Contributed by
NM
, 29-Mar-1998)
Ref
Expression
Assertion
preq1
⊢
A
=
B
→
A
C
=
B
C
Proof
Step
Hyp
Ref
Expression
1
sneq
⊢
A
=
B
→
A
=
B
2
1
uneq1d
⊢
A
=
B
→
A
∪
C
=
B
∪
C
3
df-pr
⊢
A
C
=
A
∪
C
4
df-pr
⊢
B
C
=
B
∪
C
5
2
3
4
3eqtr4g
⊢
A
=
B
→
A
C
=
B
C