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Theorem rusbcALT 41538
 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 {𝑥𝑥𝑥} ∉ V

Proof of Theorem rusbcALT
StepHypRef Expression
1 pm5.19 391 . . 3 ¬ ({𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥} ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥})
2 sbcnel12g 4308 . . . 4 ({𝑥𝑥𝑥} ∈ V → ([{𝑥𝑥𝑥} / 𝑥]𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥))
3 sbc8g 3704 . . . 4 ({𝑥𝑥𝑥} ∈ V → ([{𝑥𝑥𝑥} / 𝑥]𝑥𝑥 ↔ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
4 df-nel 3056 . . . . 5 ({𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ ¬ {𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥)
5 csbvarg 4328 . . . . . . 7 ({𝑥𝑥𝑥} ∈ V → {𝑥𝑥𝑥} / 𝑥𝑥 = {𝑥𝑥𝑥})
65, 5eleq12d 2846 . . . . . 6 ({𝑥𝑥𝑥} ∈ V → ({𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
76notbid 321 . . . . 5 ({𝑥𝑥𝑥} ∈ V → (¬ {𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
84, 7syl5bb 286 . . . 4 ({𝑥𝑥𝑥} ∈ V → ({𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
92, 3, 83bitr3d 312 . . 3 ({𝑥𝑥𝑥} ∈ V → ({𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥} ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
101, 9mto 200 . 2 ¬ {𝑥𝑥𝑥} ∈ V
11 df-nel 3056 . 2 ({𝑥𝑥𝑥} ∉ V ↔ ¬ {𝑥𝑥𝑥} ∈ V)
1210, 11mpbir 234 1 {𝑥𝑥𝑥} ∉ V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∈ wcel 2111  {cab 2735   ∉ wnel 3055  Vcvv 3409  [wsbc 3696  ⦋csb 3805 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-nel 3056  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-nul 4226 This theorem is referenced by: (None)
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