Theorem ralndv1 47842

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 9555	. . 3 ⊢ ¬ 𝑥 ∈ 𝑥
2	1	pm2.21i 120	. 2 ⊢ (𝑥 ∈ 𝑥 → V ∈ 𝑥)
3	2	rgen 3081	1 ⊢ ∀𝑥 ∈ 𝑥 V ∈ 𝑥

Syntax hints:   ∈ wcel 2143  ∀wral 3079  Vcvv 3455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-sep 5257  ax-reg 9550

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-ral 3080

This theorem is referenced by: (None)