Theorem altopth1

Description: Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012)

	Ref	Expression
Assertion	altopth1	$$⊢A\in V\to \left(⟪A,B⟫=⟪C,D⟫\to A=C\right)$$

Proof

Step	Hyp	Ref	Expression
1		altopthsn	$$⊢⟪A,B⟫=⟪C,D⟫\leftrightarrow \left(\left\{A\right\}=\left\{C\right\}\wedge \left\{B\right\}=\left\{D\right\}\right)$$
2		sneqrg	$$⊢A\in V\to \left(\left\{A\right\}=\left\{C\right\}\to A=C\right)$$
3	2	adantrd	$$⊢A\in V\to \left(\left(\left\{A\right\}=\left\{C\right\}\wedge \left\{B\right\}=\left\{D\right\}\right)\to A=C\right)$$
4	1 3	biimtrid	$$⊢A\in V\to \left(⟪A,B⟫=⟪C,D⟫\to A=C\right)$$