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Theorem rusbcALT 37462
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 pm5.19 373 . . 3 ¬ ({𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥} ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥})
2 sbcnel12g 3933 . . . 4 ({𝑥𝑥𝑥} ∈ V → ([{𝑥𝑥𝑥} / 𝑥]𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥))
3 sbc8g 3406 . . . 4 ({𝑥𝑥𝑥} ∈ V → ([{𝑥𝑥𝑥} / 𝑥]𝑥𝑥 ↔ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
4 df-nel 2779 . . . . 5 ({𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ ¬ {𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥)
5 csbvarg 3951 . . . . . . 7 ({𝑥𝑥𝑥} ∈ V → {𝑥𝑥𝑥} / 𝑥𝑥 = {𝑥𝑥𝑥})
65, 5eleq12d 2678 . . . . . 6 ({𝑥𝑥𝑥} ∈ V → ({𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
76notbid 306 . . . . 5 ({𝑥𝑥𝑥} ∈ V → (¬ {𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
84, 7syl5bb 270 . . . 4 ({𝑥𝑥𝑥} ∈ V → ({𝑥𝑥𝑥} / 𝑥𝑥{𝑥𝑥𝑥} / 𝑥𝑥 ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
92, 3, 83bitr3d 296 . . 3 ({𝑥𝑥𝑥} ∈ V → ({𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥} ↔ ¬ {𝑥𝑥𝑥} ∈ {𝑥𝑥𝑥}))
101, 9mto 186 . 2 ¬ {𝑥𝑥𝑥} ∈ V
11 df-nel 2779 . 2 ({𝑥𝑥𝑥} ∉ V ↔ ¬ {𝑥𝑥𝑥} ∈ V)
1210, 11mpbir 219 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wcel 1976  {cab 2592  wnel 2777  Vcvv 3169  [wsbc 3398  csb 3495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-nel 2779  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-nul 3871
This theorem is referenced by: (None)
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