Theorem ralndv1 46366

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 9590	. . 3 ⊢ ¬ 𝑥 ∈ 𝑥
2	1	pm2.21i 119	. 2 ⊢ (𝑥 ∈ 𝑥 → V ∈ 𝑥)
3	2	rgen 3057	1 ⊢ ∀𝑥 ∈ 𝑥 V ∈ 𝑥

Syntax hints:   ∈ wcel 2098  ∀wral 3055  Vcvv 3468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-pr 5420  ax-reg 9586

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-v 3470  df-un 3948  df-sn 4624  df-pr 4626

This theorem is referenced by: (None)