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Theorem eldifvsn 4806
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 4795 . 2 (𝐴 ∈ (V ∖ {𝐵}) ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
2 elex 3482 . . 3 (𝐴𝑉𝐴 ∈ V)
32biantrurd 531 . 2 (𝐴𝑉 → (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵)))
41, 3bitr4id 289 1 (𝐴𝑉 → (𝐴 ∈ (V ∖ {𝐵}) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2099  wne 2930  Vcvv 3462  cdif 3944  {csn 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-v 3464  df-dif 3950  df-sn 4634
This theorem is referenced by:  cnvimadfsn  8186
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