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Theorem isconcl6ab 26207
Description: Two distinct non-parallel lines intersect in one and only point. Proposition 4 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypotheses
Ref Expression
isconcl5a.1  |-  L  =  (PLines `  F )
isconcl5a.2  |-  P  =  (PPoints `  F )
isconcl5a.3  |-  ( ph  ->  F  e. Ig )
isconcl5a.4  |-  ( ph  -> 
L1  e.  L )
isconcl5a.5  |-  ( ph  ->  L 2  e.  L
)
isconcl5a.6  |-  ( ph  -> 
L1  =/=  L 2
)
isconcl6ab.1  |-  ( ph  ->  ( L1  i^i  L 2 )  =/=  (/) )
Assertion
Ref Expression
isconcl6ab  |-  ( ph  ->  E! p ( p  e.  L1  /\  p  e.  L 2 ) )
Distinct variable groups:    p, L1    p, L 2    ph, p
Allowed substitution hints:    P( p)    F( p)    L( p)

Proof of Theorem isconcl6ab
StepHypRef Expression
1 isconcl5a.1 . . 3  |-  L  =  (PLines `  F )
2 isconcl5a.2 . . 3  |-  P  =  (PPoints `  F )
3 isconcl5a.3 . . 3  |-  ( ph  ->  F  e. Ig )
4 isconcl5a.4 . . 3  |-  ( ph  -> 
L1  e.  L )
5 isconcl5a.5 . . 3  |-  ( ph  ->  L 2  e.  L
)
6 isconcl5a.6 . . 3  |-  ( ph  -> 
L1  =/=  L 2
)
71, 2, 3, 4, 5, 6isconcl5ab 26205 . 2  |-  ( ph  ->  E* p ( p  e.  L1  /\  p  e.  L 2 ) )
8 df-mo 2161 . . 3  |-  ( E* p ( p  e.  L1  /\  p  e.  L 2 )  <->  ( E. p ( p  e.  L1  /\  p  e.  L 2 )  ->  E! p ( p  e.  L1  /\  p  e.  L 2 ) ) )
9 isconcl6ab.1 . . . . 5  |-  ( ph  ->  ( L1  i^i  L 2 )  =/=  (/) )
10 n0 3477 . . . . . 6  |-  ( (
L1  i^i  L 2
)  =/=  (/)  <->  E. p  p  e.  ( L1  i^i  L 2 ) )
11 elin 3371 . . . . . . . . 9  |-  ( p  e.  ( L1  i^i  L 2 )  <->  ( p  e.  L1  /\  p  e.  L 2 ) )
1211biimpi 186 . . . . . . . 8  |-  ( p  e.  ( L1  i^i  L 2 )  ->  (
p  e.  L1  /\  p  e.  L 2 ) )
1312eximi 1566 . . . . . . 7  |-  ( E. p  p  e.  (
L1  i^i  L 2
)  ->  E. p
( p  e.  L1  /\  p  e.  L 2
) )
1413a1d 22 . . . . . 6  |-  ( E. p  p  e.  (
L1  i^i  L 2
)  ->  ( ph  ->  E. p ( p  e.  L1  /\  p  e.  L 2 ) ) )
1510, 14sylbi 187 . . . . 5  |-  ( (
L1  i^i  L 2
)  =/=  (/)  ->  ( ph  ->  E. p ( p  e.  L1  /\  p  e.  L 2 ) ) )
169, 15mpcom 32 . . . 4  |-  ( ph  ->  E. p ( p  e.  L1  /\  p  e.  L 2 ) )
1716imim1i 54 . . 3  |-  ( ( E. p ( p  e.  L1  /\  p  e.  L 2 )  ->  E! p ( p  e.  L1  /\  p  e.  L 2 ) )  -> 
( ph  ->  E! p
( p  e.  L1  /\  p  e.  L 2
) ) )
188, 17sylbi 187 . 2  |-  ( E* p ( p  e.  L1  /\  p  e.  L 2 )  ->  ( ph  ->  E! p ( p  e.  L1  /\  p  e.  L 2 ) ) )
197, 18mpcom 32 1  |-  ( ph  ->  E! p ( p  e.  L1  /\  p  e.  L 2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   E!weu 2156   E*wmo 2157    =/= wne 2459    i^i cin 3164   (/)c0 3468   ` cfv 5271  PPointscpoints 26159  PLinescplines 26161  Igcig 26163
This theorem is referenced by:  isconcl7a  26208
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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