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Theorem minel 3831
Description: A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)
Assertion
Ref Expression
minel  |-  ( ( A  e.  B  /\  ( C  i^i  B )  =  (/) )  ->  -.  A  e.  C )

Proof of Theorem minel
StepHypRef Expression
1 inelcm 3830 . . . . 5  |-  ( ( A  e.  C  /\  A  e.  B )  ->  ( C  i^i  B
)  =/=  (/) )
21necon2bi 2665 . . . 4  |-  ( ( C  i^i  B )  =  (/)  ->  -.  ( A  e.  C  /\  A  e.  B )
)
3 imnan 428 . . . 4  |-  ( ( A  e.  C  ->  -.  A  e.  B
)  <->  -.  ( A  e.  C  /\  A  e.  B ) )
42, 3sylibr 217 . . 3  |-  ( ( C  i^i  B )  =  (/)  ->  ( A  e.  C  ->  -.  A  e.  B )
)
54con2d 120 . 2  |-  ( ( C  i^i  B )  =  (/)  ->  ( A  e.  B  ->  -.  A  e.  C )
)
65impcom 436 1  |-  ( ( A  e.  B  /\  ( C  i^i  B )  =  (/) )  ->  -.  A  e.  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 375    = wceq 1454    e. wcel 1897    i^i cin 3414   (/)c0 3742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-v 3058  df-dif 3418  df-in 3422  df-nul 3743
This theorem is referenced by:  peano5  6742  fnsuppres  6968  domunfican  7869  unwdomg  8124  dfac5  8584  ccatval2  12758  mreexexlem2d  15599  hauspwpwf1  21050
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