Theorem ralndv1 47020

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 9665	. . 3 ⊢ ¬ 𝑥 ∈ 𝑥
2	1	pm2.21i 119	. 2 ⊢ (𝑥 ∈ 𝑥 → V ∈ 𝑥)
3	2	rgen 3069	1 ⊢ ∀𝑥 ∈ 𝑥 V ∈ 𝑥

Syntax hints:   ∈ wcel 2108  ∀wral 3067  Vcvv 3488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-pr 5447  ax-reg 9661

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-v 3490  df-un 3981  df-sn 4649  df-pr 4651

This theorem is referenced by: (None)