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Theorem elhaltdp 26170
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 )
elhaltdp.1  |-  ( ph  ->  I  e. Ig )
Assertion
Ref Expression
elhaltdp  |-  ( ph  ->  A. l  e.  L  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  l  /\  y  e.  l
) )
Distinct variable groups:    x, l,
y, L    P, l, x, y
Allowed substitution hints:    ph( x, y, l)    I( x, y, l)

Proof of Theorem elhaltdp
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elhaltdp.1 . 2  |-  ( ph  ->  I  e. Ig )
2 isig.1 . . . 4  |-  P  =  (PPoints `  I )
3 isig.2 . . . 4  |-  L  =  (PLines `  I )
42, 3bisig0 26165 . . 3  |-  ( I  e. Ig 
<->  ( I  e.  _V  /\  ( A. l  e.  L  l  C_  P  /\  A. x  e.  P  A. y  e.  P  ( x  =/=  y  ->  E! l  e.  L  ( x  e.  l  /\  y  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  E. z  e.  P  ( (
x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  A. l  e.  L  -.  ( x  e.  l  /\  y  e.  l  /\  z  e.  l
) ) ) )
5 simp23 990 . . 3  |-  ( ( I  e.  _V  /\  ( A. l  e.  L  l  C_  P  /\  A. x  e.  P  A. y  e.  P  (
x  =/=  y  ->  E! l  e.  L  ( x  e.  l  /\  y  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  E. z  e.  P  ( (
x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  A. l  e.  L  -.  ( x  e.  l  /\  y  e.  l  /\  z  e.  l
) ) )  ->  A. l  e.  L  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  l  /\  y  e.  l
) )
64, 5sylbi 187 . 2  |-  ( I  e. Ig  ->  A. l  e.  L  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  l  /\  y  e.  l ) )
71, 6syl 15 1  |-  ( ph  ->  A. l  e.  L  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  l  /\  y  e.  l
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   E!wreu 2558   _Vcvv 2801    C_ wss 3165   ` cfv 5271  PPointscpoints 26159  PLinescplines 26161  Igcig 26163
This theorem is referenced by:  elhaltdp2  26171
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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