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Theorem neiopne 25154
 Description: If an intersection is not empty, its operands are not empty. (Contributed by FL, 27-Apr-2008.)
Assertion
Ref Expression
neiopne

Proof of Theorem neiopne
StepHypRef Expression
1 ineq1 3376 . . . . 5
2 incom 3374 . . . . 5
3 eqtr 2313 . . . . . 6
4 in0 3493 . . . . . 6
53, 4syl6eq 2344 . . . . 5
61, 2, 5sylancl 643 . . . 4
7 ineq2 3377 . . . . 5
8 in0 3493 . . . . 5
97, 8syl6eq 2344 . . . 4
106, 9jaoi 368 . . 3
1110necon3ai 2499 . 2
12 neanior 2544 . 2
1311, 12sylibr 203 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wo 357   wa 358   wceq 1632   wne 2459   cin 3164  c0 3468 This theorem is referenced by:  hdrmp  25809 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
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