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Theorem isig1a2 26063
Description: A line is a set of points. This axiom is not needed. (Let's recall that the incidence relation can be formalized as an abstract relation. And that the belonging relationship is only an interpretation.) However Wayne Aitken adds this axiom to his system and I will follow him. The definitions below will take advantage of it. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
Hypotheses
Ref Expression
isig.1  |-  P  =  (PPoints `  I )
isig.2  |-  L  =  (PLines `  I )
isig1a2.1  |-  ( ph  ->  I  e. Ig )
Assertion
Ref Expression
isig1a2  |-  ( ph  ->  A. l  e.  L  l  C_  P )
Distinct variable groups:    L, l    P, l
Allowed substitution hints:    ph( l)    I(
l)

Proof of Theorem isig1a2
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isig1a2.1 . 2  |-  ( ph  ->  I  e. Ig )
2 isig.1 . . . 4  |-  P  =  (PPoints `  I )
3 isig.2 . . . 4  |-  L  =  (PLines `  I )
42, 3bisig0 26062 . . 3  |-  ( I  e. Ig 
<->  ( I  e.  _V  /\  ( A. l  e.  L  l  C_  P  /\  A. x  e.  P  A. y  e.  P  ( x  =/=  y  ->  E! l  e.  L  ( x  e.  l  /\  y  e.  l
) )  /\  A. l  e.  L  E. x  e.  P  E. y  e.  P  (
x  =/=  y  /\  x  e.  l  /\  y  e.  l )
)  /\  E. x  e.  P  E. y  e.  P  E. z  e.  P  ( (
x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  A. l  e.  L  -.  ( x  e.  l  /\  y  e.  l  /\  z  e.  l
) ) ) )
5 simp21 988 . . 3  |-  ( ( I  e.  _V  /\  ( A. l  e.  L  l  C_  P  /\  A. x  e.  P  A. y  e.  P  (
x  =/=  y  ->  E! l  e.  L  ( x  e.  l  /\  y  e.  l
) )  /\  A. l  e.  L  E. x  e.  P  E. y  e.  P  (
x  =/=  y  /\  x  e.  l  /\  y  e.  l )
)  /\  E. x  e.  P  E. y  e.  P  E. z  e.  P  ( (
x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  A. l  e.  L  -.  ( x  e.  l  /\  y  e.  l  /\  z  e.  l
) ) )  ->  A. l  e.  L  l  C_  P )
64, 5sylbi 187 . 2  |-  ( I  e. Ig  ->  A. l  e.  L  l  C_  P )
71, 6syl 15 1  |-  ( ph  ->  A. l  e.  L  l  C_  P )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   E!wreu 2545   _Vcvv 2788    C_ wss 3152   ` cfv 5255  PPointscpoints 26056  PLinescplines 26058  Igcig 26060
This theorem is referenced by:  isig12  26064
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-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-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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