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Theorem isconcl7a 26208
Description: Two distinct non-parallel lines intersect in one and only point. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypotheses
Ref Expression
isconcl7a.1  |-  L  =  (PLines `  G )
isconcl7a.3  |-  ( ph  ->  G  e. Ig )
isconcl7a.4  |-  ( ph  -> 
L1  e.  L )
isconcl7a.5  |-  ( ph  ->  L 2  e.  L
)
isconcl7a.6  |-  ( ph  -> 
L1  =/=  L 2
)
isconcl7a.7  |-  ( ph  ->  X  e.  ( L1  i^i  L 2 ) )
Assertion
Ref Expression
isconcl7a  |-  ( ph  ->  ( L1  i^i  L 2 )  =  { X } )

Proof of Theorem isconcl7a
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isconcl7a.1 . . . . . 6  |-  L  =  (PLines `  G )
2 eqid 2296 . . . . . 6  |-  (PPoints `  G
)  =  (PPoints `  G
)
3 isconcl7a.3 . . . . . 6  |-  ( ph  ->  G  e. Ig )
4 isconcl7a.4 . . . . . 6  |-  ( ph  -> 
L1  e.  L )
5 isconcl7a.5 . . . . . 6  |-  ( ph  ->  L 2  e.  L
)
6 isconcl7a.6 . . . . . 6  |-  ( ph  -> 
L1  =/=  L 2
)
7 isconcl7a.7 . . . . . . 7  |-  ( ph  ->  X  e.  ( L1  i^i  L 2 ) )
8 ne0i 3474 . . . . . . 7  |-  ( X  e.  ( L1  i^i  L 2 )  ->  ( L1  i^i  L 2 )  =/=  (/) )
97, 8syl 15 . . . . . 6  |-  ( ph  ->  ( L1  i^i  L 2 )  =/=  (/) )
101, 2, 3, 4, 5, 6, 9isconcl6ab 26207 . . . . 5  |-  ( ph  ->  E! x ( x  e.  L1  /\  x  e.  L 2 ) )
11 elin 3371 . . . . . 6  |-  ( x  e.  ( L1  i^i  L 2 )  <->  ( x  e.  L1  /\  x  e.  L 2 ) )
1211eubii 2165 . . . . 5  |-  ( E! x  x  e.  (
L1  i^i  L 2
)  <->  E! x ( x  e.  L1  /\  x  e.  L 2 ) )
1310, 12sylibr 203 . . . 4  |-  ( ph  ->  E! x  x  e.  ( L1  i^i  L 2 ) )
14 eleq1 2356 . . . . . . . 8  |-  ( x  =  X  ->  (
x  e.  ( L1  i^i  L 2 )  <->  X  e.  ( L1  i^i  L 2
) ) )
1514rexsng 3686 . . . . . . 7  |-  ( X  e.  ( L1  i^i  L 2 )  ->  ( E. x  e.  { X } x  e.  ( L1  i^i  L 2 )  <->  X  e.  ( L1  i^i  L 2 ) ) )
167, 15syl 15 . . . . . 6  |-  ( ph  ->  ( E. x  e. 
{ X } x  e.  ( L1  i^i  L 2 )  <->  X  e.  ( L1  i^i  L 2
) ) )
177, 16mpbird 223 . . . . 5  |-  ( ph  ->  E. x  e.  { X } x  e.  (
L1  i^i  L 2
) )
18 exancom 1576 . . . . . 6  |-  ( E. x ( x  e.  ( L1  i^i  L 2 )  /\  x  e.  { X } )  <->  E. x ( x  e. 
{ X }  /\  x  e.  ( L1  i^i  L 2 ) ) )
19 df-rex 2562 . . . . . 6  |-  ( E. x  e.  { X } x  e.  ( L1  i^i  L 2 )  <->  E. x ( x  e. 
{ X }  /\  x  e.  ( L1  i^i  L 2 ) ) )
2018, 19bitr4i 243 . . . . 5  |-  ( E. x ( x  e.  ( L1  i^i  L 2 )  /\  x  e.  { X } )  <->  E. x  e.  { X } x  e.  ( L1  i^i  L 2 )
)
2117, 20sylibr 203 . . . 4  |-  ( ph  ->  E. x ( x  e.  ( L1  i^i  L 2 )  /\  x  e.  { X } ) )
22 eupicka 2220 . . . 4  |-  ( ( E! x  x  e.  ( L1  i^i  L 2 )  /\  E. x ( x  e.  ( L1  i^i  L 2 )  /\  x  e.  { X } ) )  ->  A. x
( x  e.  (
L1  i^i  L 2
)  ->  x  e.  { X } ) )
2313, 21, 22syl2anc 642 . . 3  |-  ( ph  ->  A. x ( x  e.  ( L1  i^i  L 2 )  ->  x  e.  { X } ) )
2414ralsng 3685 . . . . . 6  |-  ( X  e.  ( L1  i^i  L 2 )  ->  ( A. x  e.  { X } x  e.  ( L1  i^i  L 2 )  <->  X  e.  ( L1  i^i  L 2 ) ) )
257, 24syl 15 . . . . 5  |-  ( ph  ->  ( A. x  e. 
{ X } x  e.  ( L1  i^i  L 2 )  <->  X  e.  ( L1  i^i  L 2
) ) )
267, 25mpbird 223 . . . 4  |-  ( ph  ->  A. x  e.  { X } x  e.  (
L1  i^i  L 2
) )
27 df-ral 2561 . . . 4  |-  ( A. x  e.  { X } x  e.  ( L1  i^i  L 2 )  <->  A. x ( x  e. 
{ X }  ->  x  e.  ( L1  i^i  L 2 ) ) )
2826, 27sylib 188 . . 3  |-  ( ph  ->  A. x ( x  e.  { X }  ->  x  e.  ( L1  i^i  L 2 ) ) )
29 albiim 1601 . . 3  |-  ( A. x ( x  e.  ( L1  i^i  L 2 )  <->  x  e.  { X } )  <->  ( A. x ( x  e.  ( L1  i^i  L 2 )  ->  x  e.  { X } )  /\  A. x ( x  e.  { X }  ->  x  e.  (
L1  i^i  L 2
) ) ) )
3023, 28, 29sylanbrc 645 . 2  |-  ( ph  ->  A. x ( x  e.  ( L1  i^i  L 2 )  <->  x  e.  { X } ) )
31 dfcleq 2290 . 2  |-  ( (
L1  i^i  L 2
)  =  { X } 
<-> 
A. x ( x  e.  ( L1  i^i  L 2 )  <->  x  e.  { X } ) )
3230, 31sylibr 203 1  |-  ( ph  ->  ( L1  i^i  L 2 )  =  { X } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   E!weu 2156    =/= wne 2459   A.wral 2556   E.wrex 2557    i^i cin 3164   (/)c0 3468   {csn 3653   ` cfv 5271  PPointscpoints 26159  PLinescplines 26161  Igcig 26163
This theorem is referenced by:  lppotos  26247
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-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-riota 6320  df-ig2 26164  df-li 26180
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