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Theorem minel 3882
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

Proof of Theorem minel
StepHypRef Expression
1 inelcm 3881 . . . . 5
21necon2bi 2694 . . . 4
3 imnan 422 . . . 4
42, 3sylibr 212 . . 3
54con2d 115 . 2
65impcom 430 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  i^icin 3474   c0 3784
This theorem is referenced by:  fnsuppresOLD  6131  peano5  6723  fnsuppres  6946  domunfican  7813  unwdomg  8031  dfac5  8530  ccatval2  12596  mreexexlem2d  15042  hauspwpwf1  20488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-in 3482  df-nul 3785
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