Theorem ruALT 9641

Description: Alternate proof of ru 3789, simplified using (indirectly) the Axiom of Regularity ax-reg 9630. (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 5321	. . 3 ⊢ ¬ V ∈ V
2	1	nelir 3047	. 2 ⊢ V ∉ V
3		ruv 9640	. . 3 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V
4		neleq1 3050	. . 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 1537  {cab 2712   ∉ wnel 3044  Vcvv 3478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-reg 9630

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nel 3045  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634

This theorem is referenced by: (None)