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Theorem pxysxy 26245
Description: Points between  X and  Y belong to the segment XY. (For my private use only. Don't use.) (Contributed by FL, 17-Jul-2016.)
Hypotheses
Ref Expression
sgplpte.1  |-  P  =  (PPoints `  G )
sgplpte.3  |-  S  =  ( seg `  G
)
sgplpte.4  |-  ( ph  ->  G  e. Ibg )
sgplpte.5  |-  ( ph  ->  X  e.  P )
pxysxy.1  |-  B  =  (btw `  G )
pxysxy.2  |-  ( ph  ->  Y  e.  P )
pxysxy.3  |-  ( ph  ->  X  =/=  Y )
Assertion
Ref Expression
pxysxy  |-  ( ph  ->  ( ( X B Y )  i^i  P
)  C_  ( X S Y ) )

Proof of Theorem pxysxy
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elin 3371 . . . 4  |-  ( z  e.  ( ( X B Y )  i^i 
P )  <->  ( z  e.  ( X B Y )  /\  z  e.  P ) )
2 simp1 955 . . . . . . . 8  |-  ( (
ph  /\  z  e.  P  /\  z  e.  ( X B Y ) )  ->  ph )
3 simp2 956 . . . . . . . 8  |-  ( (
ph  /\  z  e.  P  /\  z  e.  ( X B Y ) )  ->  z  e.  P )
4 3mix1 1124 . . . . . . . . 9  |-  ( z  e.  ( X B Y )  ->  (
z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y )
)
543ad2ant3 978 . . . . . . . 8  |-  ( (
ph  /\  z  e.  P  /\  z  e.  ( X B Y ) )  ->  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) )
6 sgplpte.1 . . . . . . . . . 10  |-  P  =  (PPoints `  G )
7 sgplpte.3 . . . . . . . . . 10  |-  S  =  ( seg `  G
)
8 sgplpte.4 . . . . . . . . . 10  |-  ( ph  ->  G  e. Ibg )
9 sgplpte.5 . . . . . . . . . 10  |-  ( ph  ->  X  e.  P )
10 pxysxy.1 . . . . . . . . . 10  |-  B  =  (btw `  G )
11 pxysxy.2 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  P )
12 pxysxy.3 . . . . . . . . . 10  |-  ( ph  ->  X  =/=  Y )
136, 7, 8, 9, 10, 11, 12sgplpte21d 26239 . . . . . . . . 9  |-  ( ph  ->  ( ( z  e.  P  /\  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) )  -> 
z  e.  ( X S Y ) ) )
1413imp 418 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  P  /\  (
z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y )
) )  ->  z  e.  ( X S Y ) )
152, 3, 5, 14syl12anc 1180 . . . . . . 7  |-  ( (
ph  /\  z  e.  P  /\  z  e.  ( X B Y ) )  ->  z  e.  ( X S Y ) )
16153exp 1150 . . . . . 6  |-  ( ph  ->  ( z  e.  P  ->  ( z  e.  ( X B Y )  ->  z  e.  ( X S Y ) ) ) )
1716com13 74 . . . . 5  |-  ( z  e.  ( X B Y )  ->  (
z  e.  P  -> 
( ph  ->  z  e.  ( X S Y ) ) ) )
1817imp 418 . . . 4  |-  ( ( z  e.  ( X B Y )  /\  z  e.  P )  ->  ( ph  ->  z  e.  ( X S Y ) ) )
191, 18sylbi 187 . . 3  |-  ( z  e.  ( ( X B Y )  i^i 
P )  ->  ( ph  ->  z  e.  ( X S Y ) ) )
2019com12 27 . 2  |-  ( ph  ->  ( z  e.  ( ( X B Y )  i^i  P )  ->  z  e.  ( X S Y ) ) )
2120ssrdv 3198 1  |-  ( ph  ->  ( ( X B Y )  i^i  P
)  C_  ( X S Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    i^i cin 3164    C_ wss 3165   ` cfv 5271  (class class class)co 5874  PPointscpoints 26159  btwcbtw 26209  Ibgcibg 26210   segcseg 26233
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  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-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-seg2 26234
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