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Theorem lineval12a 26084
Description: The line passing through two distinct points  A and  B is a set of points . (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypotheses
Ref Expression
lineval12a.1  |-  P  =  (PPoints `  G )
lineval12a.3  |-  M  =  ( line `  G
)
lineval12a.4  |-  ( ph  ->  G  e. Ig )
lineval12a.5  |-  ( ph  ->  A  e.  P )
lineval12a.6  |-  ( ph  ->  B  e.  P )
Assertion
Ref Expression
lineval12a  |-  ( ph  ->  ( A M B )  C_  P )

Proof of Theorem lineval12a
StepHypRef Expression
1 lineval12a.1 . . . . 5  |-  P  =  (PPoints `  G )
2 lineval12a.3 . . . . 5  |-  M  =  ( line `  G
)
3 lineval12a.4 . . . . 5  |-  ( ph  ->  G  e. Ig )
4 lineval12a.6 . . . . 5  |-  ( ph  ->  B  e.  P )
51, 2, 3, 4lineval3a 26083 . . . 4  |-  ( ph  ->  ( B M B )  =  { B } )
64snssd 3760 . . . 4  |-  ( ph  ->  { B }  C_  P )
75, 6eqsstrd 3212 . . 3  |-  ( ph  ->  ( B M B )  C_  P )
8 oveq1 5865 . . . 4  |-  ( A  =  B  ->  ( A M B )  =  ( B M B ) )
98sseq1d 3205 . . 3  |-  ( A  =  B  ->  (
( A M B )  C_  P  <->  ( B M B )  C_  P
) )
107, 9syl5ibr 212 . 2  |-  ( A  =  B  ->  ( ph  ->  ( A M B )  C_  P
) )
11 eqid 2283 . . . 4  |-  (PLines `  G )  =  (PLines `  G )
123adantl 452 . . . 4  |-  ( ( A  =/=  B  /\  ph )  ->  G  e. Ig )
13 lineval12a.5 . . . . . 6  |-  ( ph  ->  A  e.  P )
1413adantl 452 . . . . 5  |-  ( ( A  =/=  B  /\  ph )  ->  A  e.  P )
154adantl 452 . . . . 5  |-  ( ( A  =/=  B  /\  ph )  ->  B  e.  P )
16 simpl 443 . . . . 5  |-  ( ( A  =/=  B  /\  ph )  ->  A  =/=  B )
171, 11, 2, 12, 14, 15, 16lineval12 26081 . . . 4  |-  ( ( A  =/=  B  /\  ph )  ->  ( A M B )  e.  (PLines `  G ) )
181, 11, 12, 17isig12 26064 . . 3  |-  ( ( A  =/=  B  /\  ph )  ->  ( A M B )  C_  P
)
1918ex 423 . 2  |-  ( A  =/=  B  ->  ( ph  ->  ( A M B )  C_  P
) )
2010, 19pm2.61ine 2522 1  |-  ( ph  ->  ( A M B )  C_  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   {csn 3640   ` cfv 5255  (class class class)co 5858  PPointscpoints 26056  PLinescplines 26058  Igcig 26060   linecline 26076
This theorem is referenced by:  lineval5a  26088  lineval6a  26089
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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-ig2 26061  df-li 26077
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