Theorem altopthg

Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012)

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

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		sneqbg	$$⊢A\in V\to \left(\left\{A\right\}=\left\{C\right\}\leftrightarrow A=C\right)$$
3		sneqbg	$$⊢B\in W\to \left(\left\{B\right\}=\left\{D\right\}\leftrightarrow B=D\right)$$
4	2 3	bi2anan9	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\left(\left\{A\right\}=\left\{C\right\}\wedge \left\{B\right\}=\left\{D\right\}\right)\leftrightarrow \left(A=C\wedge B=D\right)\right)$$
5	1 4	syl5bb	$$⊢\left(A\in V\wedge B\in W\right)\to \left(⟪A,B⟫=⟪C,D⟫\leftrightarrow \left(A=C\wedge B=D\right)\right)$$