Theorem ruALT 9523

Description: Alternate proof of ru 3740, simplified using (indirectly) the Axiom of Regularity ax-reg 9509. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ruALT	⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Proof of Theorem ruALT

Step	Hyp	Ref	Expression
1		vprc 5262	. . 3 ⊢ ¬ V ∈ V
2	1	nelir 3040	. 2 ⊢ V ∉ V
3		ruv 9522	. . 3 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V
4		neleq1 3043	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} = V → ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ V ∉ V))
5	3, 4	ax-mp 5	. 2 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ V ∉ V)
6	2, 5	mpbir 231	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Syntax hints:   ↔ wb 206   = wceq 1542  {cab 2715   ∉ wnel 3037  Vcvv 3442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-reg 9509

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nel 3038  df-v 3444

This theorem is referenced by: (None)