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Theorem eldifvsn 4703
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 elex 3489 . . 3 (𝐴𝑉𝐴 ∈ V)
21biantrurd 536 . 2 (𝐴𝑉 → (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵)))
3 eldifsn 4692 . 2 (𝐴 ∈ (V ∖ {𝐵}) ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
42, 3syl6rbbr 293 1 (𝐴𝑉 → (𝐴 ∈ (V ∖ {𝐵}) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2115  wne 3007  Vcvv 3471  cdif 3907  {csn 4540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-ne 3008  df-v 3473  df-dif 3913  df-sn 4541
This theorem is referenced by:  cnvimadfsn  7814
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