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Theorem minel 3455
 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 3454 . . . . 5
21necon2bi 2382 . . . 4
3 imnan 680 . . . 4
42, 3sylibr 133 . . 3
54con2d 614 . 2
65impcom 124 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wceq 1335   wcel 2128   cin 3101  c0 3394 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-v 2714  df-dif 3104  df-in 3108  df-nul 3395 This theorem is referenced by:  unfidisj  6863  hashunlem  10671
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