Theorem ruALT 9556

Description: Alternate proof of ru 3751, simplified using (indirectly) the Axiom of Regularity ax-reg 9545. (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 5270	. . 3 ⊢ ¬ V ∈ V
2	1	nelir 3032	. 2 ⊢ V ∉ V
3		ruv 9555	. . 3 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V
4		neleq1 3035	. . 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 1540  {cab 2707   ∉ wnel 3029  Vcvv 3447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-pr 5387  ax-reg 9545

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nel 3030  df-ral 3045  df-rex 3054  df-v 3449  df-un 3919  df-sn 4590  df-pr 4592

This theorem is referenced by: (None)