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Theorem clel4 2816
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
clel4.1 𝐵 ∈ V
Assertion
Ref Expression
clel4 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem clel4
StepHypRef Expression
1 clel4.1 . . 3 𝐵 ∈ V
2 eleq2 2201 . . 3 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
31, 2ceqsalv 2711 . 2 (∀𝑥(𝑥 = 𝐵𝐴𝑥) ↔ 𝐴𝐵)
43bicomi 131 1 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1329   = wceq 1331  wcel 1480  Vcvv 2681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-v 2683
This theorem is referenced by:  intpr  3798
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