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Theorem inteximm 4133
Description: The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
inteximm (∃𝑥 𝑥𝐴 𝐴 ∈ V)
Distinct variable group:   𝑥,𝐴

Proof of Theorem inteximm
StepHypRef Expression
1 intss1 3844 . . 3 (𝑥𝐴 𝐴𝑥)
2 vex 2733 . . . 4 𝑥 ∈ V
32ssex 4124 . . 3 ( 𝐴𝑥 𝐴 ∈ V)
41, 3syl 14 . 2 (𝑥𝐴 𝐴 ∈ V)
54exlimiv 1591 1 (∃𝑥 𝑥𝐴 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1485  wcel 2141  Vcvv 2730  wss 3121   cint 3829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4105
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-int 3830
This theorem is referenced by:  intexabim  4136  iinexgm  4138  onintonm  4499  elfi2  6945  elfir  6946  fifo  6953
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