Theorem ifpprsnss

Description: An unordered pair is a singleton or a subset of itself. This theorem is helpful to convert theorems about walks in arbitrary graphs into theorems about walks in pseudographs. (Contributed by AV, 27-Feb-2021)

	Ref	Expression
Assertion	ifpprsnss	$$⊢P=\left\{A,B\right\}\to if-\left(A=B,P=\left\{A\right\},\left\{A,B\right\}\subseteq P\right)$$

Proof

Step	Hyp	Ref	Expression
1		preq2	$$⊢B=A\to \left\{A,B\right\}=\left\{A,A\right\}$$
2		dfsn2	$$⊢\left\{A\right\}=\left\{A,A\right\}$$
3	1 2	eqtr4di	$$⊢B=A\to \left\{A,B\right\}=\left\{A\right\}$$
4	3	eqcoms	$$⊢A=B\to \left\{A,B\right\}=\left\{A\right\}$$
5	4	eqeq2d	$$⊢A=B\to \left(P=\left\{A,B\right\}\leftrightarrow P=\left\{A\right\}\right)$$
6	5	biimpac	$$⊢\left(P=\left\{A,B\right\}\wedge A=B\right)\to P=\left\{A\right\}$$
7		eqimss2	$$⊢P=\left\{A,B\right\}\to \left\{A,B\right\}\subseteq P$$
8	7	adantr	$$⊢\left(P=\left\{A,B\right\}\wedge \neg A=B\right)\to \left\{A,B\right\}\subseteq P$$
9	6 8	ifpimpda	$$⊢P=\left\{A,B\right\}\to if-\left(A=B,P=\left\{A\right\},\left\{A,B\right\}\subseteq P\right)$$