Theorem rusbcALT 38640

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		pm5.19 375	. . 3 ⊢ ¬ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥})
2		sbcnel12g 3985	. . . 4 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ([{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥]𝑥 ∉ 𝑥 ↔ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∉ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥))
3		sbc8g 3443	. . . 4 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ([{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥]𝑥 ∉ 𝑥 ↔ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥}))
4		df-nel 2898	. . . . 5 ⊢ (⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∉ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ↔ ¬ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∈ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥)
5		csbvarg 4003	. . . . . . 7 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 = {𝑥 ∣ 𝑥 ∉ 𝑥})
6	5, 5	eleq12d 2695	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → (⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∈ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ↔ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥}))
7	6	notbid 308	. . . . 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 272	. . . 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 298	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥}))
10	1, 9	mto 188	. 2 ⊢ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V
11		df-nel 2898	. 2 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V)
12	10, 11	mpbir 221	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Syntax hints:  ¬ wn 3   ↔ wb 196   ∈ wcel 1990  {cab 2608   ∉ wnel 2897  Vcvv 3200  [wsbc 3435  ⦋csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)