Theorem opeqpr

Description: Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008) (Avoid depending on this detail.)

	Ref	Expression
Hypotheses	opeqpr.1	⊢ 𝐴 ∈ V
	opeqpr.2	⊢ 𝐵 ∈ V
	opeqpr.3	⊢ 𝐶 ∈ V
	opeqpr.4	⊢ 𝐷 ∈ V
Assertion	opeqpr	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = { 𝐶 , 𝐷 } ↔ ( ( 𝐶 = { 𝐴 } ∧ 𝐷 = { 𝐴 , 𝐵 } ) ∨ ( 𝐶 = { 𝐴 , 𝐵 } ∧ 𝐷 = { 𝐴 } ) ) )

Proof

Step	Hyp	Ref	Expression
1		opeqpr.1	⊢ 𝐴 ∈ V
2		opeqpr.2	⊢ 𝐵 ∈ V
3		opeqpr.3	⊢ 𝐶 ∈ V
4		opeqpr.4	⊢ 𝐷 ∈ V
5		eqcom	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = { 𝐶 , 𝐷 } ↔ { 𝐶 , 𝐷 } = ⟨ 𝐴 , 𝐵 ⟩ )
6	1 2	dfop	⊢ ⟨ 𝐴 , 𝐵 ⟩ = { { 𝐴 } , { 𝐴 , 𝐵 } }
7	6	eqeq2i	⊢ ( { 𝐶 , 𝐷 } = ⟨ 𝐴 , 𝐵 ⟩ ↔ { 𝐶 , 𝐷 } = { { 𝐴 } , { 𝐴 , 𝐵 } } )
8		snex	⊢ { 𝐴 } ∈ V
9		prex	⊢ { 𝐴 , 𝐵 } ∈ V
10	3 4 8 9	preq12b	⊢ ( { 𝐶 , 𝐷 } = { { 𝐴 } , { 𝐴 , 𝐵 } } ↔ ( ( 𝐶 = { 𝐴 } ∧ 𝐷 = { 𝐴 , 𝐵 } ) ∨ ( 𝐶 = { 𝐴 , 𝐵 } ∧ 𝐷 = { 𝐴 } ) ) )
11	5 7 10	3bitri	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = { 𝐶 , 𝐷 } ↔ ( ( 𝐶 = { 𝐴 } ∧ 𝐷 = { 𝐴 , 𝐵 } ) ∨ ( 𝐶 = { 𝐴 , 𝐵 } ∧ 𝐷 = { 𝐴 } ) ) )