Theorem rusbcALT 45421

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		pm5.19 391	. . 3 ⊢ ¬ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥})
2		sbcnel12g 4372	. . . 4 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ([{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥]𝑥 ∉ 𝑥 ↔ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∉ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥))
3		sbc8g 3747	. . . 4 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ([{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥]𝑥 ∉ 𝑥 ↔ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥}))
4		df-nel 3063	. . . . 5 ⊢ (⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∉ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ↔ ¬ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∈ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥)
5		csbvarg 4392	. . . . . . 7 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 = {𝑥 ∣ 𝑥 ∉ 𝑥})
6	5, 5	eleq12d 2855	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → (⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ∈ ⦋{𝑥 ∣ 𝑥 ∉ 𝑥} / 𝑥⦌𝑥 ↔ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥}))
7	6	notbid 321	. . . . 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 286	. . . 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 312	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V → ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∣ 𝑥 ∉ 𝑥}))
10	1, 9	mto 200	. 2 ⊢ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V
11		df-nel 3063	. 2 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V)
12	10, 11	mpbir 234	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∈ wcel 2145  {cab 2739   ∉ wnel 3062  Vcvv 3451  [wsbc 3739  ⦋csb 3847

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-nel 3063  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-nul 4280

This theorem is used by: (None)