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Theorem rusbcALT 40648
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 pm5.19 388 . . 3 ¬ ({𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥} ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥})
2 sbcnel12g 4360 . . . 4 ({𝑥𝑥𝑥} ∈ V → ([{𝑥𝑥𝑥} / 𝑥]𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥))
3 sbc8g 3777 . . . 4 ({𝑥𝑥𝑥} ∈ V → ([{𝑥𝑥𝑥} / 𝑥]𝑥𝑥 ↔ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
4 df-nel 3121 . . . . 5 ({𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ ¬ {𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥)
5 csbvarg 4380 . . . . . . 7 ({𝑥𝑥𝑥} ∈ V → {𝑥𝑥𝑥} / 𝑥𝑥 = {𝑥𝑥𝑥})
65, 5eleq12d 2904 . . . . . 6 ({𝑥𝑥𝑥} ∈ V → ({𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
76notbid 319 . . . . 5 ({𝑥𝑥𝑥} ∈ V → (¬ {𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
84, 7syl5bb 284 . . . 4 ({𝑥𝑥𝑥} ∈ V → ({𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
92, 3, 83bitr3d 310 . . 3 ({𝑥𝑥𝑥} ∈ V → ({𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥} ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
101, 9mto 198 . 2 ¬ {𝑥𝑥𝑥} ∈ V
11 df-nel 3121 . 2 ({𝑥𝑥𝑥} ∉ V ↔ ¬ {𝑥𝑥𝑥} ∈ V)
1210, 11mpbir 232 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wcel 2105  {cab 2796  wnel 3120  Vcvv 3492  [wsbc 3769  csb 3880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-nel 3121  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-nul 4289
This theorem is referenced by: (None)
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