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Theorem rusbcALT 38957
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 pm5.19 374 . . 3 ¬ ({𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥} ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥})
2 sbcnel12g 4018 . . . 4 ({𝑥𝑥𝑥} ∈ V → ([{𝑥𝑥𝑥} / 𝑥]𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥))
3 sbc8g 3476 . . . 4 ({𝑥𝑥𝑥} ∈ V → ([{𝑥𝑥𝑥} / 𝑥]𝑥𝑥 ↔ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
4 df-nel 2927 . . . . 5 ({𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ ¬ {𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥)
5 csbvarg 4036 . . . . . . 7 ({𝑥𝑥𝑥} ∈ V → {𝑥𝑥𝑥} / 𝑥𝑥 = {𝑥𝑥𝑥})
65, 5eleq12d 2724 . . . . . 6 ({𝑥𝑥𝑥} ∈ V → ({𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
76notbid 307 . . . . 5 ({𝑥𝑥𝑥} ∈ V → (¬ {𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
84, 7syl5bb 272 . . . 4 ({𝑥𝑥𝑥} ∈ V → ({𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
92, 3, 83bitr3d 298 . . 3 ({𝑥𝑥𝑥} ∈ V → ({𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥} ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
101, 9mto 188 . 2 ¬ {𝑥𝑥𝑥} ∈ V
11 df-nel 2927 . 2 ({𝑥𝑥𝑥} ∉ V ↔ ¬ {𝑥𝑥𝑥} ∈ V)
1210, 11mpbir 221 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wcel 2030  {cab 2637  wnel 2926  Vcvv 3231  [wsbc 3468  csb 3566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-nel 2927  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-nul 3949
This theorem is referenced by: (None)
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