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Theorem ruOLD 3789
Description: Obsolete version of ru 3788 as of 20-Jun-2025. (Contributed by NM, 7-Aug-1994.) Remove use of ax-13 2374. (Revised by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ruOLD {𝑥𝑥𝑥} ∉ V

Proof of Theorem ruOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pm5.19 386 . . . . . 6 ¬ (𝑦𝑦 ↔ ¬ 𝑦𝑦)
2 eleq1w 2821 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
3 df-nel 3044 . . . . . . . . 9 (𝑥𝑥 ↔ ¬ 𝑥𝑥)
4 id 22 . . . . . . . . . . 11 (𝑥 = 𝑦𝑥 = 𝑦)
54, 4eleq12d 2832 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
65notbid 318 . . . . . . . . 9 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
73, 6bitrid 283 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑥 ↔ ¬ 𝑦𝑦))
82, 7bibi12d 345 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥𝑦𝑥𝑥) ↔ (𝑦𝑦 ↔ ¬ 𝑦𝑦)))
98spvv 1993 . . . . . 6 (∀𝑥(𝑥𝑦𝑥𝑥) → (𝑦𝑦 ↔ ¬ 𝑦𝑦))
101, 9mto 197 . . . . 5 ¬ ∀𝑥(𝑥𝑦𝑥𝑥)
11 eqabb 2878 . . . . 5 (𝑦 = {𝑥𝑥𝑥} ↔ ∀𝑥(𝑥𝑦𝑥𝑥))
1210, 11mtbir 323 . . . 4 ¬ 𝑦 = {𝑥𝑥𝑥}
1312nex 1796 . . 3 ¬ ∃𝑦 𝑦 = {𝑥𝑥𝑥}
14 isset 3491 . . 3 ({𝑥𝑥𝑥} ∈ V ↔ ∃𝑦 𝑦 = {𝑥𝑥𝑥})
1513, 14mtbir 323 . 2 ¬ {𝑥𝑥𝑥} ∈ V
1615nelir 3046 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wal 1534   = wceq 1536  wex 1775  wcel 2105  {cab 2711  wnel 3043  Vcvv 3477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nel 3044  df-v 3479
This theorem is referenced by: (None)
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