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Theorem clel4 3594
Description: Alternate definition of membership in 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 . 2 𝐵 ∈ V
2 clel4g 3593 . 2 (𝐵 ∈ V → (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥)))
31, 2ax-mp 5 1 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1537   = wceq 1539  wcel 2106  Vcvv 3432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816
This theorem is referenced by:  intprOLD  4914
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