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Theorem eldifvsn 4463
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 3364 . . 3 (𝐴𝑉𝐴 ∈ V)
21biantrurd 522 . 2 (𝐴𝑉 → (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵)))
3 eldifsn 4453 . 2 (𝐴 ∈ (V ∖ {𝐵}) ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
42, 3syl6rbbr 279 1 (𝐴𝑉 → (𝐴 ∈ (V ∖ {𝐵}) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wcel 2145  wne 2943  Vcvv 3351  cdif 3720  {csn 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3726  df-sn 4317
This theorem is referenced by:  cnvimadfsn  7453
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