Theorem prclaxpr

Description: A class that is closed under the pairing operation models the Axiom of Pairing ax-pr . Lemma II.2.4(4) of Kunen2 p. 111. (Contributed by Eric Schmidt, 29-Sep-2025)

	Ref	Expression
Assertion	prclaxpr	⊢ ( ∀ 𝑥 ∈ 𝑀 ∀ 𝑦 ∈ 𝑀 { 𝑥 , 𝑦 } ∈ 𝑀 → ∀ 𝑥 ∈ 𝑀 ∀ 𝑦 ∈ 𝑀 ∃ 𝑧 ∈ 𝑀 ∀ 𝑤 ∈ 𝑀 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) )

Proof

Step	Hyp	Ref	Expression
1		vex	⊢ 𝑤 ∈ V
2	1	elpr	⊢ ( 𝑤 ∈ { 𝑥 , 𝑦 } ↔ ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) )
3	2	biimpri	⊢ ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ { 𝑥 , 𝑦 } )
4	3	rgenw	⊢ ∀ 𝑤 ∈ 𝑀 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ { 𝑥 , 𝑦 } )
5		eleq2	⊢ ( 𝑧 = { 𝑥 , 𝑦 } → ( 𝑤 ∈ 𝑧 ↔ 𝑤 ∈ { 𝑥 , 𝑦 } ) )
6	5	imbi2d	⊢ ( 𝑧 = { 𝑥 , 𝑦 } → ( ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) ↔ ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ { 𝑥 , 𝑦 } ) ) )
7	6	ralbidv	⊢ ( 𝑧 = { 𝑥 , 𝑦 } → ( ∀ 𝑤 ∈ 𝑀 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) ↔ ∀ 𝑤 ∈ 𝑀 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ { 𝑥 , 𝑦 } ) ) )
8	7	rspcev	⊢ ( ( { 𝑥 , 𝑦 } ∈ 𝑀 ∧ ∀ 𝑤 ∈ 𝑀 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ { 𝑥 , 𝑦 } ) ) → ∃ 𝑧 ∈ 𝑀 ∀ 𝑤 ∈ 𝑀 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) )
9	4 8	mpan2	⊢ ( { 𝑥 , 𝑦 } ∈ 𝑀 → ∃ 𝑧 ∈ 𝑀 ∀ 𝑤 ∈ 𝑀 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) )
10	9	2ralimi	⊢ ( ∀ 𝑥 ∈ 𝑀 ∀ 𝑦 ∈ 𝑀 { 𝑥 , 𝑦 } ∈ 𝑀 → ∀ 𝑥 ∈ 𝑀 ∀ 𝑦 ∈ 𝑀 ∃ 𝑧 ∈ 𝑀 ∀ 𝑤 ∈ 𝑀 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) )