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Theorem aline 25486
Description: A line is not empty. (Contributed by FL, 10-Aug-2016.)
Hypotheses
Ref Expression
alne.1  |-  ( ph  ->  I  e. Ig )
alne.2  |-  L  =  (PLines `  I )
alne.3  |-  ( ph  ->  M  e.  L )
Assertion
Ref Expression
aline  |-  ( ph  ->  M  =/=  (/) )

Proof of Theorem aline
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . 4  |-  (PPoints `  I
)  =  (PPoints `  I
)
2 alne.2 . . . 4  |-  L  =  (PLines `  I )
3 alne.1 . . . 4  |-  ( ph  ->  I  e. Ig )
4 alne.3 . . . 4  |-  ( ph  ->  M  e.  L )
51, 2, 3, 4elhalop2 25481 . . 3  |-  ( ph  ->  E. x  e.  (PPoints `  I ) x  e.  M )
6 rexex 2602 . . 3  |-  ( E. x  e.  (PPoints `  I
) x  e.  M  ->  E. x  x  e.  M )
75, 6syl 15 . 2  |-  ( ph  ->  E. x  x  e.  M )
8 n0 3464 . 2  |-  ( M  =/=  (/)  <->  E. x  x  e.  M )
97, 8sylibr 203 1  |-  ( ph  ->  M  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   (/)c0 3455   ` cfv 5255  PPointscpoints 25468  PLinescplines 25470  Igcig 25472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ig2 25473
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