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Theorem gltpntl2 26073
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 )
gltpntl2.1  |-  ( ph  ->  I  e. Ig )
gltpntl2.2  |-  ( ph  ->  M  e.  L )
Assertion
Ref Expression
gltpntl2  |-  ( ph  ->  E. x  x  e.  ( P  \  M
) )
Distinct variable groups:    x, M    x, P
Allowed substitution hints:    ph( x)    I( x)    L( x)

Proof of Theorem gltpntl2
StepHypRef Expression
1 isig.1 . . 3  |-  P  =  (PPoints `  I )
2 isig.2 . . 3  |-  L  =  (PLines `  I )
3 gltpntl2.1 . . 3  |-  ( ph  ->  I  e. Ig )
4 gltpntl2.2 . . 3  |-  ( ph  ->  M  e.  L )
51, 2, 3, 4gltpntl 26072 . 2  |-  ( ph  ->  E. x  e.  P  x  e/  M )
6 df-rex 2549 . . 3  |-  ( E. x  e.  P  x  e/  M  <->  E. x
( x  e.  P  /\  x  e/  M ) )
7 df-nel 2449 . . . . 5  |-  ( x  e/  M  <->  -.  x  e.  M )
8 eldif 3162 . . . . . 6  |-  ( x  e.  ( P  \  M )  <->  ( x  e.  P  /\  -.  x  e.  M ) )
98biimpri 197 . . . . 5  |-  ( ( x  e.  P  /\  -.  x  e.  M
)  ->  x  e.  ( P  \  M ) )
107, 9sylan2b 461 . . . 4  |-  ( ( x  e.  P  /\  x  e/  M )  ->  x  e.  ( P  \  M ) )
1110eximi 1563 . . 3  |-  ( E. x ( x  e.  P  /\  x  e/  M )  ->  E. x  x  e.  ( P  \  M ) )
126, 11sylbi 187 . 2  |-  ( E. x  e.  P  x  e/  M  ->  E. x  x  e.  ( P  \  M ) )
135, 12syl 15 1  |-  ( ph  ->  E. x  x  e.  ( P  \  M
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    e/ wnel 2447   E.wrex 2544    \ cdif 3149   ` cfv 5255  PPointscpoints 26056  PLinescplines 26058  Igcig 26060
This theorem is referenced by:  hpd  26169  bhp3  26177
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ig2 26061
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