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Theorem intexab 4953
 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 4101 . 2 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
2 intex 4951 . 2 ({𝑥𝜑} ≠ ∅ ↔ {𝑥𝜑} ∈ V)
31, 2bitr3i 266 1 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  ∃wex 1852   ∈ wcel 2145  {cab 2757   ≠ wne 2943  Vcvv 3351  ∅c0 4063  ∩ cint 4611 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-nul 4064  df-int 4612 This theorem is referenced by:  intexrab  4954  tcmin  8781  cfval  9271  efgval  18337  relintabex  38413  rclexi  38448  rtrclex  38450  trclexi  38453  rtrclexi  38454
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