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 = { 𝐴 , 𝐵 } → if- ( 𝐴 = 𝐵 , 𝑃 = { 𝐴 } , { 𝐴 , 𝐵 } ⊆ 𝑃 ) )

Proof

Step	Hyp	Ref	Expression
1		preq2	⊢ ( 𝐵 = 𝐴 → { 𝐴 , 𝐵 } = { 𝐴 , 𝐴 } )
2		dfsn2	⊢ { 𝐴 } = { 𝐴 , 𝐴 }
3	1 2	eqtr4di	⊢ ( 𝐵 = 𝐴 → { 𝐴 , 𝐵 } = { 𝐴 } )
4	3	eqcoms	⊢ ( 𝐴 = 𝐵 → { 𝐴 , 𝐵 } = { 𝐴 } )
5	4	eqeq2d	⊢ ( 𝐴 = 𝐵 → ( 𝑃 = { 𝐴 , 𝐵 } ↔ 𝑃 = { 𝐴 } ) )
6	5	biimpac	⊢ ( ( 𝑃 = { 𝐴 , 𝐵 } ∧ 𝐴 = 𝐵 ) → 𝑃 = { 𝐴 } )
7		eqimss2	⊢ ( 𝑃 = { 𝐴 , 𝐵 } → { 𝐴 , 𝐵 } ⊆ 𝑃 )
8	7	adantr	⊢ ( ( 𝑃 = { 𝐴 , 𝐵 } ∧ ¬ 𝐴 = 𝐵 ) → { 𝐴 , 𝐵 } ⊆ 𝑃 )
9	6 8	ifpimpda	⊢ ( 𝑃 = { 𝐴 , 𝐵 } → if- ( 𝐴 = 𝐵 , 𝑃 = { 𝐴 } , { 𝐴 , 𝐵 } ⊆ 𝑃 ) )