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Theorem lineval12 26081
Description: The line passing through two distinct points  A and  B is a line. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypotheses
Ref Expression
lineval2.1  |-  P  =  (PPoints `  G )
lineval2.2  |-  L  =  (PLines `  G )
lineval2.3  |-  M  =  ( line `  G
)
lineval2.4  |-  ( ph  ->  G  e. Ig )
lineval12.5  |-  ( ph  ->  A  e.  P )
lineval12.6  |-  ( ph  ->  B  e.  P )
lineval12.7  |-  ( ph  ->  A  =/=  B )
Assertion
Ref Expression
lineval12  |-  ( ph  ->  ( A M B )  e.  L )

Proof of Theorem lineval12
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lineval2.1 . . 3  |-  P  =  (PPoints `  G )
2 lineval2.2 . . 3  |-  L  =  (PLines `  G )
3 lineval2.3 . . 3  |-  M  =  ( line `  G
)
4 lineval2.4 . . 3  |-  ( ph  ->  G  e. Ig )
5 lineval12.5 . . 3  |-  ( ph  ->  A  e.  P )
6 lineval12.6 . . 3  |-  ( ph  ->  B  e.  P )
7 lineval12.7 . . 3  |-  ( ph  ->  A  =/=  B )
81, 2, 3, 4, 5, 6, 7lineval222 26079 . 2  |-  ( ph  ->  ( A M B )  =  ( iota_ x  e.  L ( A  e.  x  /\  B  e.  x ) ) )
91, 2, 4, 5, 6, 7isig2a2 26066 . . 3  |-  ( ph  ->  E! x  e.  L  ( A  e.  x  /\  B  e.  x
) )
10 riotacl 6319 . . 3  |-  ( E! x  e.  L  ( A  e.  x  /\  B  e.  x )  ->  ( iota_ x  e.  L
( A  e.  x  /\  B  e.  x
) )  e.  L
)
119, 10syl 15 . 2  |-  ( ph  ->  ( iota_ x  e.  L
( A  e.  x  /\  B  e.  x
) )  e.  L
)
128, 11eqeltrd 2357 1  |-  ( ph  ->  ( A M B )  e.  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E!wreu 2545   ` cfv 5255  (class class class)co 5858   iota_crio 6297  PPointscpoints 26056  PLinescplines 26058  Igcig 26060   linecline 26076
This theorem is referenced by:  lineval12a  26084  lineval5a  26088  lineval6a  26089  lppotos  26144
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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