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Theorem class2seteq 3944
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 2583 . 2 (𝐴𝑉𝐴 ∈ V)
2 ax-1 5 . . . . 5 (𝐴 ∈ V → (𝑥𝐴𝐴 ∈ V))
32ralrimiv 2408 . . . 4 (𝐴 ∈ V → ∀𝑥𝐴 𝐴 ∈ V)
4 rabid2 2503 . . . 4 (𝐴 = {𝑥𝐴𝐴 ∈ V} ↔ ∀𝑥𝐴 𝐴 ∈ V)
53, 4sylibr 141 . . 3 (𝐴 ∈ V → 𝐴 = {𝑥𝐴𝐴 ∈ V})
65eqcomd 2061 . 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 1259  wcel 1409  wral 2323  {crab 2327  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-ral 2328  df-rab 2332  df-v 2576
This theorem is referenced by: (None)
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