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Theorem isconcl6a 25502
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
)
isconcl6a.1  |-  ( ph  ->  ( L1  i^i  L 2 )  =/=  (/) )
Assertion
Ref Expression
isconcl6a  |-  ( ph  ->  E! p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
Distinct variable groups:    p, L1    p, L 2    P, p    ph, p
Allowed substitution hints:    F( p)    L( p)

Proof of Theorem isconcl6a
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, 6isconcl5a 25500 . 2  |-  ( ph  ->  E* p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
8 df-mo 2149 . . 3  |-  ( E* p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  <->  ( E. p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  ->  E! p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) ) )
9 isconcl6a.1 . . . . 5  |-  ( ph  ->  ( L1  i^i  L 2 )  =/=  (/) )
10 n0 3465 . . . . . 6  |-  ( (
L1  i^i  L 2
)  =/=  (/)  <->  E. p  p  e.  ( L1  i^i  L 2 ) )
112, 1, 3, 4isig12 25463 . . . . . . . 8  |-  ( ph  -> 
L1  C_  P )
12 elin 3359 . . . . . . . . . . 11  |-  ( p  e.  ( L1  i^i  L 2 )  <->  ( p  e.  L1  /\  p  e.  L 2 ) )
13 ssel 3175 . . . . . . . . . . . . . . 15  |-  ( L1  C_  P  ->  ( p  e.  L1  ->  p  e.  P ) )
1413impcom 421 . . . . . . . . . . . . . 14  |-  ( ( p  e.  L1  /\  L1  C_  P
)  ->  p  e.  P )
15143adant2 979 . . . . . . . . . . . . 13  |-  ( ( p  e.  L1  /\  p  e.  L 2  /\  L1  C_  P
)  ->  p  e.  P )
16 simp1 960 . . . . . . . . . . . . 13  |-  ( ( p  e.  L1  /\  p  e.  L 2  /\  L1  C_  P
)  ->  p  e.  L1 )
17 simp2 961 . . . . . . . . . . . . 13  |-  ( ( p  e.  L1  /\  p  e.  L 2  /\  L1  C_  P
)  ->  p  e.  L 2 )
1815, 16, 173jca 1137 . . . . . . . . . . . 12  |-  ( ( p  e.  L1  /\  p  e.  L 2  /\  L1  C_  P
)  ->  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
19183expia 1158 . . . . . . . . . . 11  |-  ( ( p  e.  L1  /\  p  e.  L 2 )  -> 
( L1  C_  P  -> 
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) ) )
2012, 19sylbi 189 . . . . . . . . . 10  |-  ( p  e.  ( L1  i^i  L 2 )  ->  ( L1  C_  P  ->  (
p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) ) )
2120com12 29 . . . . . . . . 9  |-  ( L1  C_  P  ->  ( p  e.  ( L1  i^i  L 2 )  ->  (
p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) ) )
2221eximdv 1613 . . . . . . . 8  |-  ( L1  C_  P  ->  ( E. p  p  e.  ( L1  i^i  L 2 )  ->  E. p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) ) )
2311, 22syl 17 . . . . . . 7  |-  ( ph  ->  ( E. p  p  e.  ( L1  i^i  L 2 )  ->  E. p
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) ) )
2423com12 29 . . . . . 6  |-  ( E. p  p  e.  (
L1  i^i  L 2
)  ->  ( ph  ->  E. p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) ) )
2510, 24sylbi 189 . . . . 5  |-  ( (
L1  i^i  L 2
)  =/=  (/)  ->  ( ph  ->  E. p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) ) )
269, 25mpcom 34 . . . 4  |-  ( ph  ->  E. p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
2726imim1i 56 . . 3  |-  ( ( E. p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  ->  E! p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  -> 
( ph  ->  E! p
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) ) )
288, 27sylbi 189 . 2  |-  ( E* p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  ->  ( ph  ->  E! p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) ) )
297, 28mpcom 34 1  |-  ( ph  ->  E! p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939   E.wex 1533    = wceq 1628    e. wcel 1688   E!weu 2144   E*wmo 2145    =/= wne 2447    i^i cin 3152    C_ wss 3153   (/)c0 3456   ` cfv 5221  PPointscpoints 25455  PLinescplines 25457  Igcig 25459
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-ig2 25460  df-li 25476
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