Theorem ralndv1 44637

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 9383	. . 3 ⊢ ¬ 𝑥 ∈ 𝑥
2	1	pm2.21i 119	. 2 ⊢ (𝑥 ∈ 𝑥 → V ∈ 𝑥)
3	2	rgen 3061	1 ⊢ ∀𝑥 ∈ 𝑥 V ∈ 𝑥

Syntax hints:   ∈ wcel 2101  ∀wral 3059  Vcvv 3434

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 2103  ax-9 2111  ax-10 2132  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-reg 9379

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3060  df-rex 3069  df-v 3436  df-dif 3892  df-un 3894  df-nul 4260  df-sn 4565  df-pr 4567

This theorem is referenced by: (None)