Theorem ruALT 4597

Description: Alternate proof of Russell's Paradox ru 2996, simplified using (indirectly) the Axiom of Set Induction ax-setind 4583. (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 4175	. . 3 ⊢ ¬ V ∈ V
2		df-nel 2471	. . 3 ⊢ (V ∉ V ↔ ¬ V ∈ V)
3	1, 2	mpbir 146	. 2 ⊢ V ∉ V
4		ruv 4596	. . 3 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V
5		neleq1 2474	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} = V → ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ V ∉ V))
6	4, 5	ax-mp 5	. 2 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ V ∉ V)
7	3, 6	mpbir 146	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Syntax hints:  ¬ wn 3   ↔ wb 105   = wceq 1372   ∈ wcel 2175  {cab 2190   ∉ wnel 2470  Vcvv 2771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-setind 4583

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-v 2773  df-dif 3167  df-sn 3638

This theorem is referenced by: (None)