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Theorem minel 2314
Description: A minimum element of a class has no elements in common with the class.
Assertion
Ref Expression
minel |- ((A e. B /\ (C i^i B) = (/)) -> -. A e. C)
Proof of Theorem minel
StepHypRef Expression
1 inelcm 2313 . . . . 5 |- ((A e. C /\ A e. B) -> (C i^i B) =/= (/))
21necon2bi 1604 . . . 4 |- ((C i^i B) = (/) -> -. (A e. C /\ A e. B))
3 imnan 242 . . . 4 |- ((A e. C -> -. A e. B) <-> -. (A e. C /\ A e. B))
42, 3sylibr 200 . . 3 |- ((C i^i B) = (/) -> (A e. C -> -. A e. B))
54con2d 91 . 2 |- ((C i^i B) = (/) -> (A e. B -> -. A e. C))
65impcom 351 1 |- ((A e. B /\ (C i^i B) = (/)) -> -. A e. C)
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955   i^i cin 2036  (/)c0 2270
This theorem is referenced by:  peano5 3143  aceq5 4712
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-in 2041  df-nul 2271
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