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

Theorem noelOLD 4184
 Description: Obsolete version of noel 4183 as of 3-May-2023. The empty set has no elements. Theorem 6.14 of [Quine] p. 44. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
noelOLD ¬ 𝐴 ∈ ∅

Proof of Theorem noelOLD
StepHypRef Expression
1 eldifi 3993 . . 3 (𝐴 ∈ (V ∖ V) → 𝐴 ∈ V)
2 eldifn 3994 . . 3 (𝐴 ∈ (V ∖ V) → ¬ 𝐴 ∈ V)
31, 2pm2.65i 186 . 2 ¬ 𝐴 ∈ (V ∖ V)
4 df-nul 4179 . . 3 ∅ = (V ∖ V)
54eleq2i 2857 . 2 (𝐴 ∈ ∅ ↔ 𝐴 ∈ (V ∖ V))
63, 5mtbir 315 1 ¬ 𝐴 ∈ ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 2050  Vcvv 3415   ∖ cdif 3826  ∅c0 4178 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-v 3417  df-dif 3832  df-nul 4179 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator