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Theorem intexab 5300
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 4344 . 2 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
2 intex 5298 . 2 ({𝑥𝜑} ≠ ∅ ↔ {𝑥𝜑} ∈ V)
31, 2bitr3i 277 1 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wex 1782  wcel 2107  {cab 2710  wne 2940  Vcvv 3447  c0 4286   cint 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-in 3921  df-ss 3931  df-nul 4287  df-int 4912
This theorem is referenced by:  intexrab  5301  tcmin  9685  cfval  10191  efgval  19507  relintabex  41945  rclexi  41979  rtrclex  41981  trclexi  41984  rtrclexi  41985  aiotaexb  45411
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