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Theorem gltpntl 26175
Description: Given a line, there exists a point not on this line. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
Hypotheses
Ref Expression
isig.1  |-  P  =  (PPoints `  I )
isig.2  |-  L  =  (PLines `  I )
gltpntl.1  |-  ( ph  ->  I  e. Ig )
gltpntl.2  |-  ( ph  ->  M  e.  L )
Assertion
Ref Expression
gltpntl  |-  ( ph  ->  E. x  e.  P  x  e/  M )
Distinct variable groups:    x, M    x, P
Allowed substitution hints:    ph( x)    I( x)    L( x)

Proof of Theorem gltpntl
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isig.1 . . 3  |-  P  =  (PPoints `  I )
2 isig.2 . . 3  |-  L  =  (PLines `  I )
3 gltpntl.1 . . 3  |-  ( ph  ->  I  e. Ig )
4 gltpntl.2 . . 3  |-  ( ph  ->  M  e.  L )
51, 2, 3, 4tethpnc2 26174 . 2  |-  ( ph  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  ( ( a  =/=  b  /\  b  =/=  c  /\  a  =/=  c )  /\  -.  ( a  e.  M  /\  b  e.  M  /\  c  e.  M
) ) )
6 3ianor 949 . . . . . . . 8  |-  ( -.  ( a  e.  M  /\  b  e.  M  /\  c  e.  M
)  <->  ( -.  a  e.  M  \/  -.  b  e.  M  \/  -.  c  e.  M
) )
7 df-nel 2462 . . . . . . . . . 10  |-  ( a  e/  M  <->  -.  a  e.  M )
8 neleq1 2550 . . . . . . . . . . . . . . 15  |-  ( x  =  a  ->  (
x  e/  M  <->  a  e/  M ) )
98rspcev 2897 . . . . . . . . . . . . . 14  |-  ( ( a  e.  P  /\  a  e/  M )  ->  E. x  e.  P  x  e/  M )
109ex 423 . . . . . . . . . . . . 13  |-  ( a  e.  P  ->  (
a  e/  M  ->  E. x  e.  P  x  e/  M ) )
1110adantr 451 . . . . . . . . . . . 12  |-  ( ( a  e.  P  /\  b  e.  P )  ->  ( a  e/  M  ->  E. x  e.  P  x  e/  M ) )
1211com12 27 . . . . . . . . . . 11  |-  ( a  e/  M  ->  (
( a  e.  P  /\  b  e.  P
)  ->  E. x  e.  P  x  e/  M ) )
1312a1d 22 . . . . . . . . . 10  |-  ( a  e/  M  ->  (
c  e.  P  -> 
( ( a  e.  P  /\  b  e.  P )  ->  E. x  e.  P  x  e/  M ) ) )
147, 13sylbir 204 . . . . . . . . 9  |-  ( -.  a  e.  M  -> 
( c  e.  P  ->  ( ( a  e.  P  /\  b  e.  P )  ->  E. x  e.  P  x  e/  M ) ) )
15 df-nel 2462 . . . . . . . . . 10  |-  ( b  e/  M  <->  -.  b  e.  M )
16 neleq1 2550 . . . . . . . . . . . . . . 15  |-  ( x  =  b  ->  (
x  e/  M  <->  b  e/  M ) )
1716rspcev 2897 . . . . . . . . . . . . . 14  |-  ( ( b  e.  P  /\  b  e/  M )  ->  E. x  e.  P  x  e/  M )
1817ex 423 . . . . . . . . . . . . 13  |-  ( b  e.  P  ->  (
b  e/  M  ->  E. x  e.  P  x  e/  M ) )
1918adantl 452 . . . . . . . . . . . 12  |-  ( ( a  e.  P  /\  b  e.  P )  ->  ( b  e/  M  ->  E. x  e.  P  x  e/  M ) )
2019com12 27 . . . . . . . . . . 11  |-  ( b  e/  M  ->  (
( a  e.  P  /\  b  e.  P
)  ->  E. x  e.  P  x  e/  M ) )
2120a1d 22 . . . . . . . . . 10  |-  ( b  e/  M  ->  (
c  e.  P  -> 
( ( a  e.  P  /\  b  e.  P )  ->  E. x  e.  P  x  e/  M ) ) )
2215, 21sylbir 204 . . . . . . . . 9  |-  ( -.  b  e.  M  -> 
( c  e.  P  ->  ( ( a  e.  P  /\  b  e.  P )  ->  E. x  e.  P  x  e/  M ) ) )
23 df-nel 2462 . . . . . . . . . 10  |-  ( c  e/  M  <->  -.  c  e.  M )
24 neleq1 2550 . . . . . . . . . . . . 13  |-  ( x  =  c  ->  (
x  e/  M  <->  c  e/  M ) )
2524rspcev 2897 . . . . . . . . . . . 12  |-  ( ( c  e.  P  /\  c  e/  M )  ->  E. x  e.  P  x  e/  M )
2625expcom 424 . . . . . . . . . . 11  |-  ( c  e/  M  ->  (
c  e.  P  ->  E. x  e.  P  x  e/  M ) )
2726a1dd 42 . . . . . . . . . 10  |-  ( c  e/  M  ->  (
c  e.  P  -> 
( ( a  e.  P  /\  b  e.  P )  ->  E. x  e.  P  x  e/  M ) ) )
2823, 27sylbir 204 . . . . . . . . 9  |-  ( -.  c  e.  M  -> 
( c  e.  P  ->  ( ( a  e.  P  /\  b  e.  P )  ->  E. x  e.  P  x  e/  M ) ) )
2914, 22, 283jaoi 1245 . . . . . . . 8  |-  ( ( -.  a  e.  M  \/  -.  b  e.  M  \/  -.  c  e.  M
)  ->  ( c  e.  P  ->  ( ( a  e.  P  /\  b  e.  P )  ->  E. x  e.  P  x  e/  M ) ) )
306, 29sylbi 187 . . . . . . 7  |-  ( -.  ( a  e.  M  /\  b  e.  M  /\  c  e.  M
)  ->  ( c  e.  P  ->  ( ( a  e.  P  /\  b  e.  P )  ->  E. x  e.  P  x  e/  M ) ) )
3130adantl 452 . . . . . 6  |-  ( ( ( a  =/=  b  /\  b  =/=  c  /\  a  =/=  c
)  /\  -.  (
a  e.  M  /\  b  e.  M  /\  c  e.  M )
)  ->  ( c  e.  P  ->  ( ( a  e.  P  /\  b  e.  P )  ->  E. x  e.  P  x  e/  M ) ) )
3231com12 27 . . . . 5  |-  ( c  e.  P  ->  (
( ( a  =/=  b  /\  b  =/=  c  /\  a  =/=  c )  /\  -.  ( a  e.  M  /\  b  e.  M  /\  c  e.  M
) )  ->  (
( a  e.  P  /\  b  e.  P
)  ->  E. x  e.  P  x  e/  M ) ) )
3332rexlimiv 2674 . . . 4  |-  ( E. c  e.  P  ( ( a  =/=  b  /\  b  =/=  c  /\  a  =/=  c
)  /\  -.  (
a  e.  M  /\  b  e.  M  /\  c  e.  M )
)  ->  ( (
a  e.  P  /\  b  e.  P )  ->  E. x  e.  P  x  e/  M ) )
3433com12 27 . . 3  |-  ( ( a  e.  P  /\  b  e.  P )  ->  ( E. c  e.  P  ( ( a  =/=  b  /\  b  =/=  c  /\  a  =/=  c )  /\  -.  ( a  e.  M  /\  b  e.  M  /\  c  e.  M
) )  ->  E. x  e.  P  x  e/  M ) )
3534rexlimivv 2685 . 2  |-  ( E. a  e.  P  E. b  e.  P  E. c  e.  P  (
( a  =/=  b  /\  b  =/=  c  /\  a  =/=  c
)  /\  -.  (
a  e.  M  /\  b  e.  M  /\  c  e.  M )
)  ->  E. x  e.  P  x  e/  M )
365, 35syl 15 1  |-  ( ph  ->  E. x  e.  P  x  e/  M )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    e/ wnel 2460   E.wrex 2557   ` cfv 5271  PPointscpoints 26159  PLinescplines 26161  Igcig 26163
This theorem is referenced by:  gltpntl2  26176  abhp  26276
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ig2 26164
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