Theorem rusbcALT 41946

Description: A version of Russell's paradox which is proven using proper substitution. (Contributed by Andrew Salmon, 18-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
rusbcALT	⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Proof of Theorem rusbcALT

Step	Hyp	Ref	Expression
1		pm5.19 387	. . 3 ⊢ ¬ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥})
2		sbcnel12g 4342	. . . 4 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ([{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥]𝑥 ∉ 𝑥 ↔ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∉ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥))
3		sbc8g 3719	. . . 4 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ([{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥]𝑥 ∉ 𝑥 ↔ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥}))
4		df-nel 3049	. . . . 5 ⊢ (⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∉ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ↔ ¬ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∈ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥)
5		csbvarg 4362	. . . . . . 7 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 = {𝑥 ∣ 𝑥 ∉ 𝑥})
6	5, 5	eleq12d 2833	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → (⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∈ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ↔ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥}))
7	6	notbid 317	. . . . 5 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → (¬ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∈ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥}))
8	4, 7	syl5bb 282	. . . 4 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → (⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∉ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥}))
9	2, 3, 8	3bitr3d 308	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥}))
10	1, 9	mto 196	. 2 ⊢ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V
11		df-nel 3049	. 2 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V)
12	10, 11	mpbir 230	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Syntax hints:  ¬ wn 3   ↔ wb 205   ∈ wcel 2108  {cab 2715   ∉ wnel 3048  Vcvv 3422  [wsbc 3711  ⦋csb 3828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-nel 3049  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-nul 4254

This theorem is referenced by: (None)