MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  minel Unicode version

Theorem minel 3523
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 3522 . . . . 5  |-  ( ( A  e.  C  /\  A  e.  B )  ->  ( C  i^i  B
)  =/=  (/) )
21necon2bi 2505 . . . 4  |-  ( ( C  i^i  B )  =  (/)  ->  -.  ( A  e.  C  /\  A  e.  B )
)
3 imnan 411 . . . 4  |-  ( ( A  e.  C  ->  -.  A  e.  B
)  <->  -.  ( A  e.  C  /\  A  e.  B ) )
42, 3sylibr 203 . . 3  |-  ( ( C  i^i  B )  =  (/)  ->  ( A  e.  C  ->  -.  A  e.  B )
)
54con2d 107 . 2  |-  ( ( C  i^i  B )  =  (/)  ->  ( A  e.  B  ->  -.  A  e.  C )
)
65impcom 419 1  |-  ( ( A  e.  B  /\  ( C  i^i  B )  =  (/) )  ->  -.  A  e.  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164   (/)c0 3468
This theorem is referenced by:  peano5  4695  fnsuppres  5748  domunfican  7145  unwdomg  7314  dfac5  7771  ccatval2  11448  mreexexlem2d  13563  hauspwpwf1  17698
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-in 3172  df-nul 3469
  Copyright terms: Public domain W3C validator