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Theorem intexab 5012
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 4153 . 2 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
2 intex 5010 . 2 ({𝑥𝜑} ≠ ∅ ↔ {𝑥𝜑} ∈ V)
31, 2bitr3i 269 1 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wex 1875  wcel 2157  {cab 2783  wne 2969  Vcvv 3383  c0 4113   cint 4665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-v 3385  df-dif 3770  df-in 3774  df-ss 3781  df-nul 4114  df-int 4666
This theorem is referenced by:  intexrab  5013  tcmin  8865  cfval  9355  efgval  18440  relintabex  38658  rclexi  38693  rtrclex  38695  trclexi  38698  rtrclexi  38699  aiotaexb  41926
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