Theorem wfaxpr

Description: The class of well-founded sets models the Axiom of Pairing ax-pr . Part of Corollary II.2.5 of Kunen2 p. 112. (Contributed by Eric Schmidt, 29-Sep-2025)

	Ref	Expression
Hypothesis	wfax.1	⊢ 𝑊 = ∪ ( 𝑅1 “ On )
Assertion	wfaxpr	⊢ ∀ 𝑥 ∈ 𝑊 ∀ 𝑦 ∈ 𝑊 ∃ 𝑧 ∈ 𝑊 ∀ 𝑤 ∈ 𝑊 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 )

Proof

Step	Hyp	Ref	Expression
1		wfax.1	⊢ 𝑊 = ∪ ( 𝑅1 “ On )
2		prwf	⊢ ( ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝑦 ∈ ∪ ( 𝑅1 “ On ) ) → { 𝑥 , 𝑦 } ∈ ∪ ( 𝑅1 “ On ) )
3	1	eleq2i	⊢ ( 𝑥 ∈ 𝑊 ↔ 𝑥 ∈ ∪ ( 𝑅1 “ On ) )
4	1	eleq2i	⊢ ( 𝑦 ∈ 𝑊 ↔ 𝑦 ∈ ∪ ( 𝑅1 “ On ) )
5	3 4	anbi12i	⊢ ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ↔ ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝑦 ∈ ∪ ( 𝑅1 “ On ) ) )
6	1	eleq2i	⊢ ( { 𝑥 , 𝑦 } ∈ 𝑊 ↔ { 𝑥 , 𝑦 } ∈ ∪ ( 𝑅1 “ On ) )
7	2 5 6	3imtr4i	⊢ ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) → { 𝑥 , 𝑦 } ∈ 𝑊 )
8	7	rgen2	⊢ ∀ 𝑥 ∈ 𝑊 ∀ 𝑦 ∈ 𝑊 { 𝑥 , 𝑦 } ∈ 𝑊
9		prclaxpr	⊢ ( ∀ 𝑥 ∈ 𝑊 ∀ 𝑦 ∈ 𝑊 { 𝑥 , 𝑦 } ∈ 𝑊 → ∀ 𝑥 ∈ 𝑊 ∀ 𝑦 ∈ 𝑊 ∃ 𝑧 ∈ 𝑊 ∀ 𝑤 ∈ 𝑊 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) )
10	8 9	ax-mp	⊢ ∀ 𝑥 ∈ 𝑊 ∀ 𝑦 ∈ 𝑊 ∃ 𝑧 ∈ 𝑊 ∀ 𝑤 ∈ 𝑊 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 )