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Theorem elhaltdp2 26171
Description: Every line has at least two distinct points. (For my private use only. Don't use.) (Contributed by FL, 28-Apr-2016.)
Hypotheses
Ref Expression
isig.1  |-  P  =  (PPoints `  I )
isig.2  |-  L  =  (PLines `  I )
elhaltdp2.1  |-  ( ph  ->  I  e. Ig )
elhaltdp2.2  |-  ( ph  ->  A  e.  L )
Assertion
Ref Expression
elhaltdp2  |-  ( ph  ->  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  A  /\  y  e.  A
) )
Distinct variable groups:    x, y, A    x, L, y    x, P, y
Allowed substitution hints:    ph( x, y)    I( x, y)

Proof of Theorem elhaltdp2
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 elhaltdp2.2 . 2  |-  ( ph  ->  A  e.  L )
2 isig.1 . . 3  |-  P  =  (PPoints `  I )
3 isig.2 . . 3  |-  L  =  (PLines `  I )
4 elhaltdp2.1 . . 3  |-  ( ph  ->  I  e. Ig )
52, 3, 4elhaltdp 26170 . 2  |-  ( ph  ->  A. l  e.  L  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  l  /\  y  e.  l
) )
6 eleq2 2357 . . . . 5  |-  ( l  =  A  ->  (
x  e.  l  <->  x  e.  A ) )
7 eleq2 2357 . . . . 5  |-  ( l  =  A  ->  (
y  e.  l  <->  y  e.  A ) )
86, 73anbi23d 1255 . . . 4  |-  ( l  =  A  ->  (
( x  =/=  y  /\  x  e.  l  /\  y  e.  l
)  <->  ( x  =/=  y  /\  x  e.  A  /\  y  e.  A ) ) )
982rexbidv 2599 . . 3  |-  ( l  =  A  ->  ( E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  l  /\  y  e.  l
)  <->  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  A  /\  y  e.  A
) ) )
109rspcv 2893 . 2  |-  ( A  e.  L  ->  ( A. l  e.  L  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  l  /\  y  e.  l
)  ->  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  A  /\  y  e.  A ) ) )
111, 5, 10sylc 56 1  |-  ( ph  ->  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  A  /\  y  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   ` cfv 5271  PPointscpoints 26159  PLinescplines 26161  Igcig 26163
This theorem is referenced by:  elhalop2  26172
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  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ig2 26164
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