Theorem prprvalpw

Description: The set of all proper unordered pairs over a given set V , expressed by a restricted class abstraction. (Contributed by AV, 29-Apr-2023)

	Ref	Expression
Assertion	prprvalpw	⊢ ( 𝑉 ∈ 𝑊 → ( Pairsproper ‘ 𝑉 ) = { 𝑝 ∈ 𝒫 𝑉 ∣ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) } )

Proof

Step	Hyp	Ref	Expression
1		prprval	⊢ ( 𝑉 ∈ 𝑊 → ( Pairsproper ‘ 𝑉 ) = { 𝑝 ∣ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) } )
2		prssi	⊢ ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → { 𝑎 , 𝑏 } ⊆ 𝑉 )
3		eleq1	⊢ ( 𝑝 = { 𝑎 , 𝑏 } → ( 𝑝 ∈ 𝒫 𝑉 ↔ { 𝑎 , 𝑏 } ∈ 𝒫 𝑉 ) )
4	3	adantl	⊢ ( ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) → ( 𝑝 ∈ 𝒫 𝑉 ↔ { 𝑎 , 𝑏 } ∈ 𝒫 𝑉 ) )
5		prex	⊢ { 𝑎 , 𝑏 } ∈ V
6	5	elpw	⊢ ( { 𝑎 , 𝑏 } ∈ 𝒫 𝑉 ↔ { 𝑎 , 𝑏 } ⊆ 𝑉 )
7	4 6	bitrdi	⊢ ( ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) → ( 𝑝 ∈ 𝒫 𝑉 ↔ { 𝑎 , 𝑏 } ⊆ 𝑉 ) )
8	2 7	syl5ibrcom	⊢ ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → ( ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) → 𝑝 ∈ 𝒫 𝑉 ) )
9	8	rexlimivv	⊢ ( ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) → 𝑝 ∈ 𝒫 𝑉 )
10	9	pm4.71ri	⊢ ( ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) ↔ ( 𝑝 ∈ 𝒫 𝑉 ∧ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) ) )
11	10	a1i	⊢ ( 𝑉 ∈ 𝑊 → ( ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) ↔ ( 𝑝 ∈ 𝒫 𝑉 ∧ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) ) ) )
12	11	abbidv	⊢ ( 𝑉 ∈ 𝑊 → { 𝑝 ∣ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) } = { 𝑝 ∣ ( 𝑝 ∈ 𝒫 𝑉 ∧ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) ) } )
13		df-rab	⊢ { 𝑝 ∈ 𝒫 𝑉 ∣ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) } = { 𝑝 ∣ ( 𝑝 ∈ 𝒫 𝑉 ∧ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) ) }
14	12 13	eqtr4di	⊢ ( 𝑉 ∈ 𝑊 → { 𝑝 ∣ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) } = { 𝑝 ∈ 𝒫 𝑉 ∣ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) } )
15	1 14	eqtrd	⊢ ( 𝑉 ∈ 𝑊 → ( Pairsproper ‘ 𝑉 ) = { 𝑝 ∈ 𝒫 𝑉 ∣ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑝 = { 𝑎 , 𝑏 } ) } )