Theorem opid

Description: The ordered pair <. A , A >. in Kuratowski's representation. Inference form of opidg . (Contributed by FL, 28-Dec-2011) (Proof shortened by AV, 16-Feb-2022) (Avoid depending on this detail.)

	Ref	Expression
Hypothesis	opid.1	$$⊢A\in \mathrm{V}$$
Assertion	opid	$$⊢⟨A,A⟩=\left\{\left\{A\right\}\right\}$$

Proof

Step	Hyp	Ref	Expression
1		opid.1	$$⊢A\in \mathrm{V}$$
2		opidg	$$⊢A\in \mathrm{V}\to ⟨A,A⟩=\left\{\left\{A\right\}\right\}$$
3	1 2	ax-mp	$$⊢⟨A,A⟩=\left\{\left\{A\right\}\right\}$$