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Theorem clel3 3373
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 3372 . 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 196  wa 383   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233
This theorem is referenced by:  unipr  4481  brcup  32171  brcap  32172
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