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Theorem intexab 4749
 Description: The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.)
Assertion
Ref Expression
intexab (∃𝑥𝜑 {𝑥𝜑} ∈ V)

Proof of Theorem intexab
StepHypRef Expression
1 abn0 3908 . 2 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
2 intex 4747 . 2 ({𝑥𝜑} ≠ ∅ ↔ {𝑥𝜑} ∈ V)
31, 2bitr3i 265 1 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  Vcvv 3173  ∅c0 3874  ∩ cint 4410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-int 4411 This theorem is referenced by:  intexrab  4750  tcmin  8500  cfval  8952  efgval  17953  relintabex  36906  rclexi  36941  rtrclex  36943  trclexi  36946  rtrclexi  36947
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