MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  noelOLD Structured version   Visualization version   GIF version

Theorem noelOLD 4335
Description: Obsolete version of noel 4334 as of 18-Sep-2024. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
noelOLD ¬ 𝐴 ∈ ∅

Proof of Theorem noelOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pm3.24 401 . . . . . . 7 ¬ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)
21nex 1794 . . . . . 6 ¬ ∃𝑦(𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)
3 df-clab 2706 . . . . . . 7 (𝑥 ∈ {𝑦 ∣ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)} ↔ [𝑥 / 𝑦](𝑦 ∈ V ∧ ¬ 𝑦 ∈ V))
4 spsbe 2077 . . . . . . 7 ([𝑥 / 𝑦](𝑦 ∈ V ∧ ¬ 𝑦 ∈ V) → ∃𝑦(𝑦 ∈ V ∧ ¬ 𝑦 ∈ V))
53, 4sylbi 216 . . . . . 6 (𝑥 ∈ {𝑦 ∣ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)} → ∃𝑦(𝑦 ∈ V ∧ ¬ 𝑦 ∈ V))
62, 5mto 196 . . . . 5 ¬ 𝑥 ∈ {𝑦 ∣ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)}
7 df-nul 4327 . . . . . . 7 ∅ = (V ∖ V)
8 df-dif 3952 . . . . . . 7 (V ∖ V) = {𝑦 ∣ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)}
97, 8eqtri 2756 . . . . . 6 ∅ = {𝑦 ∣ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)}
109eleq2i 2821 . . . . 5 (𝑥 ∈ ∅ ↔ 𝑥 ∈ {𝑦 ∣ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)})
116, 10mtbir 322 . . . 4 ¬ 𝑥 ∈ ∅
1211intnan 485 . . 3 ¬ (𝑥 = 𝐴𝑥 ∈ ∅)
1312nex 1794 . 2 ¬ ∃𝑥(𝑥 = 𝐴𝑥 ∈ ∅)
14 dfclel 2807 . 2 (𝐴 ∈ ∅ ↔ ∃𝑥(𝑥 = 𝐴𝑥 ∈ ∅))
1513, 14mtbir 322 1 ¬ 𝐴 ∈ ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 394   = wceq 1533  wex 1773  [wsb 2059  wcel 2098  {cab 2705  Vcvv 3473  cdif 3946  c0 4326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-dif 3952  df-nul 4327
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator