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Theorem minel 3628
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 3627 . . . . 5  |-  ( ( A  e.  C  /\  A  e.  B )  ->  ( C  i^i  B
)  =/=  (/) )
21necon2bi 2598 . . . 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 1649    e. wcel 1717    i^i cin 3264   (/)c0 3573
This theorem is referenced by:  peano5  4810  fnsuppres  5893  domunfican  7317  unwdomg  7487  dfac5  7944  ccatval2  11675  mreexexlem2d  13799  hauspwpwf1  17942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-v 2903  df-dif 3268  df-in 3272  df-nul 3574
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