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Theorem eldifvsn 4797
Description: A set is an element of the universal class excluding a singleton iff it is not the singleton element. (Contributed by AV, 7-Apr-2019.)
Assertion
Ref Expression
eldifvsn (𝐴𝑉 → (𝐴 ∈ (V ∖ {𝐵}) ↔ 𝐴𝐵))

Proof of Theorem eldifvsn
StepHypRef Expression
1 eldifsn 4786 . 2 (𝐴 ∈ (V ∖ {𝐵}) ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
2 elex 3501 . . 3 (𝐴𝑉𝐴 ∈ V)
32biantrurd 532 . 2 (𝐴𝑉 → (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵)))
41, 3bitr4id 290 1 (𝐴𝑉 → (𝐴 ∈ (V ∖ {𝐵}) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  wne 2940  Vcvv 3480  cdif 3948  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-sn 4627
This theorem is referenced by:  cnvimadfsn  8197
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