Theorem ruALT 4275

Description: Alternate proof of Russell's Paradox ru 2763, simplified using (indirectly) the Axiom of Set Induction ax-setind 4262. (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 3888	. . 3 ⊢ ¬ V ∈ V
2		df-nel 2207	. . 3 ⊢ (V ∉ V ↔ ¬ V ∈ V)
3	1, 2	mpbir 134	. 2 ⊢ V ∉ V
4		ruv 4274	. . 3 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V
5		neleq1 2301	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} = V → ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ V ∉ V))
6	4, 5	ax-mp 7	. 2 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ V ∉ V)
7	3, 6	mpbir 134	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Syntax hints:  ¬ wn 3   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {cab 2026   ∉ wnel 2205  Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-v 2559  df-dif 2920  df-sn 3381

This theorem is referenced by: (None)