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Theorem clel3 3531
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
clel3.1 𝐵 ∈ V
Assertion
Ref Expression
clel3 (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem clel3
StepHypRef Expression
1 clel3.1 . 2 𝐵 ∈ V
2 clel3g 3530 . 2 (𝐵 ∈ V → (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥)))
31, 2ax-mp 5 1 (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  wex 1875  wcel 2157  Vcvv 3385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-v 3387
This theorem is referenced by:  unipr  4641  brcup  32559  brcap  32560
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