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Theorem clel2 3639
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
clel2.1 𝐴 ∈ V
Assertion
Ref Expression
clel2 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem clel2
StepHypRef Expression
1 clel2.1 . 2 𝐴 ∈ V
2 clel2g 3638 . 2 (𝐴 ∈ V → (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵)))
31, 2ax-mp 5 1 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536   = wceq 1538  wcel 2115  Vcvv 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-nf 1786  df-cleq 2817  df-clel 2896
This theorem is referenced by:  mptelee  26692
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