Theorem vsmext 36008

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 3476	. . . 4 ⊢ (∀𝑧 ∈ V (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
2		ax-ext 2732	. . . 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 3081	. 2 ⊢ ∀𝑥 ∈ V ∀𝑦 ∈ V (∀𝑧 ∈ V (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
5		df-smext 36002	. 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 1567  ∀wral 3076  Vcvv 3450   ⊨ wsmext 36000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-smext 36002

This theorem is used by: (None)