Theorem ralndv1 47900

Description: Example for a theorem about a restricted universal quantification in which the restricting class depends on (actually is) the bound variable: All sets containing themselves contain the universal class. (Contributed by AV, 24-Jun-2023.)

Assertion

Ref	Expression
ralndv1	⊢ ∀𝑥 ∈ 𝑥 V ∈ 𝑥

Proof of Theorem ralndv1

Step	Hyp	Ref	Expression
1		elirrv 9566	. . 3 ⊢ ¬ 𝑥 ∈ 𝑥
2	1	pm2.21i 120	. 2 ⊢ (𝑥 ∈ 𝑥 → V ∈ 𝑥)
3	2	rgen 3083	1 ⊢ ∀𝑥 ∈ 𝑥 V ∈ 𝑥

This proof depends on syntax axioms:   ∈ wcel 2146  ∀wral 3081  Vcvv 3457

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-sep 5259  ax-reg 9561

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-ral 3082

This theorem is used by: (None)