Theorem ruALT 9672

Description: Alternate proof of ru 3802, simplified using (indirectly) the Axiom of Regularity ax-reg 9661. (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 5333	. . 3 ⊢ ¬ V ∈ V
2	1	nelir 3055	. 2 ⊢ V ∉ V
3		ruv 9671	. . 3 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V
4		neleq1 3058	. . 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 2717   ∉ wnel 3052  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-nel 3053  df-ral 3068  df-rex 3077  df-v 3490  df-un 3981  df-sn 4649  df-pr 4651

This theorem is referenced by: (None)