Theorem vsmext

Description: The universe is a standard model of the Axiom of Extensionality. (Contributed by BTernaryTau, 2-Oct-2026)

	Ref	Expression
Assertion	vsmext	⊢ V ⊨ Ax-Ext

Proof

Step	Hyp	Ref	Expression
1		ralv	⊢ ( ∀ 𝑧 ∈ V ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) )
2		ax-ext	⊢ ( ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) → 𝑥 = 𝑦 )
3	1 2	sylbi	⊢ ( ∀ 𝑧 ∈ V ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) → 𝑥 = 𝑦 )
4	3	rgen2w	⊢ ∀ 𝑥 ∈ V ∀ 𝑦 ∈ V ( ∀ 𝑧 ∈ V ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) → 𝑥 = 𝑦 )
5		df-smext	⊢ ( V ⊨ Ax-Ext ↔ ∀ 𝑥 ∈ V ∀ 𝑦 ∈ V ( ∀ 𝑧 ∈ V ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) → 𝑥 = 𝑦 ) )
6	4 5	mpbir	⊢ V ⊨ Ax-Ext