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Theorem class2seteq 4093
Description: Equality theorem for classes and sets . (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 2700 . 2 (𝐴𝑉𝐴 ∈ V)
2 ax-1 6 . . . . 5 (𝐴 ∈ V → (𝑥𝐴𝐴 ∈ V))
32ralrimiv 2507 . . . 4 (𝐴 ∈ V → ∀𝑥𝐴 𝐴 ∈ V)
4 rabid2 2610 . . . 4 (𝐴 = {𝑥𝐴𝐴 ∈ V} ↔ ∀𝑥𝐴 𝐴 ∈ V)
53, 4sylibr 133 . . 3 (𝐴 ∈ V → 𝐴 = {𝑥𝐴𝐴 ∈ V})
65eqcomd 2146 . 2 (𝐴 ∈ V → {𝑥𝐴𝐴 ∈ V} = 𝐴)
71, 6syl 14 1 (𝐴𝑉 → {𝑥𝐴𝐴 ∈ V} = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  wcel 1481  wral 2417  {crab 2421  Vcvv 2689
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-rab 2426  df-v 2691
This theorem is referenced by: (None)
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