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Theorem class2seteq 3713
Description: Writing a set as a class abstraction. This theorem looks artificial but was added to characterize the class abstraction whose existence is proved in class2set 5361. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
Assertion
Ref Expression
class2seteq (𝐴𝑉 → {𝑥𝐴𝐴 ∈ V} = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem class2seteq
StepHypRef Expression
1 elex 3499 . 2 (𝐴𝑉𝐴 ∈ V)
2 ax-1 6 . . 3 (𝐴 ∈ V → (𝑥𝐴𝐴 ∈ V))
32ralrimiv 3143 . 2 (𝐴 ∈ V → ∀𝑥𝐴 𝐴 ∈ V)
4 rabid2im 3467 . . 3 (∀𝑥𝐴 𝐴 ∈ V → 𝐴 = {𝑥𝐴𝐴 ∈ V})
54eqcomd 2741 . 2 (∀𝑥𝐴 𝐴 ∈ V → {𝑥𝐴𝐴 ∈ V} = 𝐴)
61, 3, 53syl 18 1 (𝐴𝑉 → {𝑥𝐴𝐴 ∈ V} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480
This theorem is referenced by: (None)
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