Theorem cnvsng

Description: Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015) (Proof shortened by BJ, 12-Feb-2022)

	Ref	Expression
Assertion	cnvsng	$$⊢\left(A\in V\wedge B\in W\right)\to {\left\{⟨A,B⟩\right\}}^{-1}=\left\{⟨B,A⟩\right\}$$

Proof

Step	Hyp	Ref	Expression
1		cnvcnvsn	$$⊢{{\left\{⟨B,A⟩\right\}}^{-1}}^{-1}={\left\{⟨A,B⟩\right\}}^{-1}$$
2		relsnopg	$$⊢\left(B\in W\wedge A\in V\right)\to \mathrm{Rel}\left\{⟨B,A⟩\right\}$$
3	2	ancoms	$$⊢\left(A\in V\wedge B\in W\right)\to \mathrm{Rel}\left\{⟨B,A⟩\right\}$$
4		dfrel2	$$⊢\mathrm{Rel}\left\{⟨B,A⟩\right\}\leftrightarrow {{\left\{⟨B,A⟩\right\}}^{-1}}^{-1}=\left\{⟨B,A⟩\right\}$$
5	3 4	sylib	$$⊢\left(A\in V\wedge B\in W\right)\to {{\left\{⟨B,A⟩\right\}}^{-1}}^{-1}=\left\{⟨B,A⟩\right\}$$
6	1 5	syl5eqr	$$⊢\left(A\in V\wedge B\in W\right)\to {\left\{⟨A,B⟩\right\}}^{-1}=\left\{⟨B,A⟩\right\}$$