Theorem vsmext 36089

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

Assertion

Ref	Expression
vsmext	⊢ V ⊨ Ax-Ext

Proof of Theorem vsmext

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ralv 3480	. . . 4 ⊢ (∀𝑧 ∈ V (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
2		ax-ext 2735	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
3	1, 2	sylbi 220	. . 3 ⊢ (∀𝑧 ∈ V (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
4	3	rgen2w 3085	. 2 ⊢ ∀𝑥 ∈ V ∀𝑦 ∈ V (∀𝑧 ∈ V (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
5		df-smext 36083	. 2 ⊢ (V ⊨ Ax-Ext ↔ ∀𝑥 ∈ V ∀𝑦 ∈ V (∀𝑧 ∈ V (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))
6	4, 5	mpbir 234	1 ⊢ V ⊨ Ax-Ext

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∀wral 3080  Vcvv 3454   ⊨ wsmext 36081

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-v 3456  df-smext 36083

This theorem is used by: (None)