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Theorem iscol3 26094
Description: Collinear points are points. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
iscola2.1  |-  ( ph  ->  G  e. Ig )
iscola3.2  |-  ( ph  ->  A  e.  (coln `  G ) )
iscola3.3  |-  P  =  (PPoints `  G )
Assertion
Ref Expression
iscol3  |-  ( ph  ->  A. x  e.  A  x  e.  P )
Distinct variable groups:    x, A    x, P
Allowed substitution hints:    ph( x)    G( x)

Proof of Theorem iscol3
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 iscola3.2 . 2  |-  ( ph  ->  A  e.  (coln `  G ) )
2 iscola2.1 . . . . 5  |-  ( ph  ->  G  e. Ig )
32, 1iscol2 26093 . . . 4  |-  ( ph  ->  ( A  e.  (coln `  G )  <->  E. l  e.  (PLines `  G ) A  C_  l ) )
4 eqid 2283 . . . . . . . . . 10  |-  (PPoints `  G
)  =  (PPoints `  G
)
5 eqid 2283 . . . . . . . . . 10  |-  (PLines `  G )  =  (PLines `  G )
62adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  l  e.  (PLines `  G ) )  ->  G  e. Ig )
7 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  l  e.  (PLines `  G ) )  ->  l  e.  (PLines `  G ) )
84, 5, 6, 7isig12 26064 . . . . . . . . 9  |-  ( (
ph  /\  l  e.  (PLines `  G ) )  ->  l  C_  (PPoints `  G ) )
9 iscola3.3 . . . . . . . . . . . 12  |-  P  =  (PPoints `  G )
10 sstr 3187 . . . . . . . . . . . 12  |-  ( ( A  C_  l  /\  l  C_  (PPoints `  G
) )  ->  A  C_  (PPoints `  G )
)
11 sseq2 3200 . . . . . . . . . . . . 13  |-  ( P  =  (PPoints `  G
)  ->  ( A  C_  P  <->  A  C_  (PPoints `  G
) ) )
12 dfss3 3170 . . . . . . . . . . . . . . 15  |-  ( A 
C_  P  <->  A. x  e.  A  x  e.  P )
1312a1i 10 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A  C_  P  <->  A. x  e.  A  x  e.  P ) )
1413biimpcd 215 . . . . . . . . . . . . 13  |-  ( A 
C_  P  ->  ( ph  ->  A. x  e.  A  x  e.  P )
)
1511, 14syl6bir 220 . . . . . . . . . . . 12  |-  ( P  =  (PPoints `  G
)  ->  ( A  C_  (PPoints `  G )  ->  ( ph  ->  A. x  e.  A  x  e.  P ) ) )
169, 10, 15mpsyl 59 . . . . . . . . . . 11  |-  ( ( A  C_  l  /\  l  C_  (PPoints `  G
) )  ->  ( ph  ->  A. x  e.  A  x  e.  P )
)
1716ex 423 . . . . . . . . . 10  |-  ( A 
C_  l  ->  (
l  C_  (PPoints `  G
)  ->  ( ph  ->  A. x  e.  A  x  e.  P )
) )
1817com3l 75 . . . . . . . . 9  |-  ( l 
C_  (PPoints `  G )  ->  ( ph  ->  ( A  C_  l  ->  A. x  e.  A  x  e.  P ) ) )
198, 18syl 15 . . . . . . . 8  |-  ( (
ph  /\  l  e.  (PLines `  G ) )  ->  ( ph  ->  ( A  C_  l  ->  A. x  e.  A  x  e.  P ) ) )
2019ex 423 . . . . . . 7  |-  ( ph  ->  ( l  e.  (PLines `  G )  ->  ( ph  ->  ( A  C_  l  ->  A. x  e.  A  x  e.  P )
) ) )
2120pm2.43a 45 . . . . . 6  |-  ( ph  ->  ( l  e.  (PLines `  G )  ->  ( A  C_  l  ->  A. x  e.  A  x  e.  P ) ) )
2221com3l 75 . . . . 5  |-  ( l  e.  (PLines `  G
)  ->  ( A  C_  l  ->  ( ph  ->  A. x  e.  A  x  e.  P )
) )
2322rexlimiv 2661 . . . 4  |-  ( E. l  e.  (PLines `  G ) A  C_  l  ->  ( ph  ->  A. x  e.  A  x  e.  P ) )
243, 23syl6bi 219 . . 3  |-  ( ph  ->  ( A  e.  (coln `  G )  ->  ( ph  ->  A. x  e.  A  x  e.  P )
) )
2524pm2.43a 45 . 2  |-  ( ph  ->  ( A  e.  (coln `  G )  ->  A. x  e.  A  x  e.  P ) )
261, 25mpd 14 1  |-  ( ph  ->  A. x  e.  A  x  e.  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   ` cfv 5255  PPointscpoints 26056  PLinescplines 26058  Igcig 26060  colnccol 26090
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-mo 2148  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-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ig2 26061  df-col 26091
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