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Theorem tethpnc2 26174
Description: There exists three non colinear points. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
Hypotheses
Ref Expression
isig.1  |-  P  =  (PPoints `  I )
isig.2  |-  L  =  (PLines `  I )
tethpnc2.1  |-  ( ph  ->  I  e. Ig )
tethpnc2.2  |-  ( ph  ->  M  e.  L )
Assertion
Ref Expression
tethpnc2  |-  ( ph  ->  E. x  e.  P  E. y  e.  P  E. z  e.  P  ( ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  -.  ( x  e.  M  /\  y  e.  M  /\  z  e.  M
) ) )
Distinct variable groups:    x, y,
z, L    x, P, y, z    ph, x, y, z
Allowed substitution hints:    I( x, y, z)    M( x, y, z)

Proof of Theorem tethpnc2
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 isig.1 . . 3  |-  P  =  (PPoints `  I )
2 isig.2 . . 3  |-  L  =  (PLines `  I )
3 tethpnc2.1 . . 3  |-  ( ph  ->  I  e. Ig )
41, 2, 3tethpnc 26173 . 2  |-  ( ph  ->  E. x  e.  P  E. y  e.  P  E. z  e.  P  ( ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  A. m  e.  L  -.  ( x  e.  m  /\  y  e.  m  /\  z  e.  m
) ) )
5 tethpnc2.2 . . . . . . 7  |-  ( ph  ->  M  e.  L )
6 eleq2 2357 . . . . . . . . . 10  |-  ( m  =  M  ->  (
x  e.  m  <->  x  e.  M ) )
7 eleq2 2357 . . . . . . . . . 10  |-  ( m  =  M  ->  (
y  e.  m  <->  y  e.  M ) )
8 eleq2 2357 . . . . . . . . . 10  |-  ( m  =  M  ->  (
z  e.  m  <->  z  e.  M ) )
96, 7, 83anbi123d 1252 . . . . . . . . 9  |-  ( m  =  M  ->  (
( x  e.  m  /\  y  e.  m  /\  z  e.  m
)  <->  ( x  e.  M  /\  y  e.  M  /\  z  e.  M ) ) )
109notbid 285 . . . . . . . 8  |-  ( m  =  M  ->  ( -.  ( x  e.  m  /\  y  e.  m  /\  z  e.  m
)  <->  -.  ( x  e.  M  /\  y  e.  M  /\  z  e.  M ) ) )
1110rspcv 2893 . . . . . . 7  |-  ( M  e.  L  ->  ( A. m  e.  L  -.  ( x  e.  m  /\  y  e.  m  /\  z  e.  m
)  ->  -.  (
x  e.  M  /\  y  e.  M  /\  z  e.  M )
) )
125, 11syl 15 . . . . . 6  |-  ( ph  ->  ( A. m  e.  L  -.  ( x  e.  m  /\  y  e.  m  /\  z  e.  m )  ->  -.  ( x  e.  M  /\  y  e.  M  /\  z  e.  M
) ) )
1312anim2d 548 . . . . 5  |-  ( ph  ->  ( ( ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  A. m  e.  L  -.  ( x  e.  m  /\  y  e.  m  /\  z  e.  m
) )  ->  (
( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z
)  /\  -.  (
x  e.  M  /\  y  e.  M  /\  z  e.  M )
) ) )
1413reximdv 2667 . . . 4  |-  ( ph  ->  ( E. z  e.  P  ( ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  A. m  e.  L  -.  ( x  e.  m  /\  y  e.  m  /\  z  e.  m
) )  ->  E. z  e.  P  ( (
x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  -.  ( x  e.  M  /\  y  e.  M  /\  z  e.  M
) ) ) )
1514reximdv 2667 . . 3  |-  ( ph  ->  ( E. y  e.  P  E. z  e.  P  ( ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  A. m  e.  L  -.  ( x  e.  m  /\  y  e.  m  /\  z  e.  m
) )  ->  E. y  e.  P  E. z  e.  P  ( (
x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  -.  ( x  e.  M  /\  y  e.  M  /\  z  e.  M
) ) ) )
1615reximdv 2667 . 2  |-  ( ph  ->  ( E. x  e.  P  E. y  e.  P  E. z  e.  P  ( ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  A. m  e.  L  -.  ( x  e.  m  /\  y  e.  m  /\  z  e.  m
) )  ->  E. x  e.  P  E. y  e.  P  E. z  e.  P  ( (
x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  -.  ( x  e.  M  /\  y  e.  M  /\  z  e.  M
) ) ) )
174, 16mpd 14 1  |-  ( ph  ->  E. x  e.  P  E. y  e.  P  E. z  e.  P  ( ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  -.  ( x  e.  M  /\  y  e.  M  /\  z  e.  M
) ) )
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   ` cfv 5271  PPointscpoints 26159  PLinescplines 26161  Igcig 26163
This theorem is referenced by:  gltpntl  26175
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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