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Theorem minel 3675
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 3674 . . . . 5  |-  ( ( A  e.  C  /\  A  e.  B )  ->  ( C  i^i  B
)  =/=  (/) )
21necon2bi 2644 . . . 4  |-  ( ( C  i^i  B )  =  (/)  ->  -.  ( A  e.  C  /\  A  e.  B )
)
3 imnan 412 . . . 4  |-  ( ( A  e.  C  ->  -.  A  e.  B
)  <->  -.  ( A  e.  C  /\  A  e.  B ) )
42, 3sylibr 204 . . 3  |-  ( ( C  i^i  B )  =  (/)  ->  ( A  e.  C  ->  -.  A  e.  B )
)
54con2d 109 . 2  |-  ( ( C  i^i  B )  =  (/)  ->  ( A  e.  B  ->  -.  A  e.  C )
)
65impcom 420 1  |-  ( ( A  e.  B  /\  ( C  i^i  B )  =  (/) )  ->  -.  A  e.  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3311   (/)c0 3620
This theorem is referenced by:  peano5  4860  fnsuppres  5944  domunfican  7371  unwdomg  7542  dfac5  7999  ccatval2  11736  mreexexlem2d  13860  hauspwpwf1  18009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-in 3319  df-nul 3621
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