Theorem ruALT 9051

Description: Alternate proof of ru 3719, simplified using (indirectly) the Axiom of Regularity ax-reg 9040. (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 5183	. . 3 ⊢ ¬ V ∈ V
2	1	nelir 3094	. 2 ⊢ V ∉ V
3		ruv 9050	. . 3 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V
4		neleq1 3096	. . 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 234	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Syntax hints:   ↔ wb 209   = wceq 1538  {cab 2776   ∉ wnel 3091  Vcvv 3441

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-reg 9040

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nel 3092  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-pr 4528

This theorem is referenced by: (None)