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Theorem intexab 5218
 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 4308 . 2 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
2 intex 5216 . 2 ({𝑥𝜑} ≠ ∅ ↔ {𝑥𝜑} ∈ V)
31, 2bitr3i 280 1 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  ∃wex 1781   ∈ wcel 2114  {cab 2800   ≠ wne 3011  Vcvv 3469  ∅c0 4265  ∩ cint 4851 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rab 3139  df-v 3471  df-dif 3911  df-in 3915  df-ss 3925  df-nul 4266  df-int 4852 This theorem is referenced by:  intexrab  5219  tcmin  9171  cfval  9658  efgval  18834  relintabex  40211  rclexi  40245  rtrclex  40247  trclexi  40250  rtrclexi  40251  aiotaexb  43585
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