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Theorem intexab 5335
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 4376 . 2 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
2 intex 5333 . 2 ({𝑥𝜑} ≠ ∅ ↔ {𝑥𝜑} ∈ V)
31, 2bitr3i 277 1 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wex 1774  wcel 2099  {cab 2705  wne 2936  Vcvv 3470  c0 4318   cint 4944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-in 3952  df-ss 3962  df-nul 4319  df-int 4945
This theorem is referenced by:  intexrab  5336  tcmin  9758  cfval  10264  efgval  19665  relintabex  43005  rclexi  43039  rtrclex  43041  trclexi  43044  rtrclexi  43045  aiotaexb  46463
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