Theorem rusbcALT 45014

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		pm5.19 389	. . 3 ⊢ ¬ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥})
2		sbcnel12g 4368	. . . 4 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ([{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥]𝑥 ∉ 𝑥 ↔ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∉ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥))
3		sbc8g 3752	. . . 4 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ([{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥]𝑥 ∉ 𝑥 ↔ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥}))
4		df-nel 3062	. . . . 5 ⊢ (⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∉ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ↔ ¬ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∈ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥)
5		csbvarg 4388	. . . . . . 7 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 = {𝑥 ∣ 𝑥 ∉ 𝑥})
6	5, 5	eleq12d 2856	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → (⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∈ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ↔ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥}))
7	6	notbid 320	. . . . 5 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → (¬ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∈ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥}))
8	4, 7	bitrid 285	. . . 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 311	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥}))
10	1, 9	mto 199	. 2 ⊢ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V
11		df-nel 3062	. 2 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V)
12	10, 11	mpbir 233	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Syntax hints:  ¬ wn 3   ↔ wb 208   ∈ wcel 2142  {cab 2740   ∉ wnel 3061  Vcvv 3454  [wsbc 3744  ⦋csb 3852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-nel 3062  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-nul 4286

This theorem is referenced by: (None)