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Theorem isig12 26064
Description: A line is a set of points. (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 )
isig12.1  |-  ( ph  ->  I  e. Ig )
isig12.2  |-  ( ph  ->  A  e.  L )
Assertion
Ref Expression
isig12  |-  ( ph  ->  A  C_  P )

Proof of Theorem isig12
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 isig12.2 . 2  |-  ( ph  ->  A  e.  L )
2 isig.1 . . 3  |-  P  =  (PPoints `  I )
3 isig.2 . . 3  |-  L  =  (PLines `  I )
4 isig12.1 . . 3  |-  ( ph  ->  I  e. Ig )
52, 3, 4isig1a2 26063 . 2  |-  ( ph  ->  A. l  e.  L  l  C_  P )
6 sseq1 3199 . . 3  |-  ( l  =  A  ->  (
l  C_  P  <->  A  C_  P
) )
76rspcva 2882 . 2  |-  ( ( A  e.  L  /\  A. l  e.  L  l 
C_  P )  ->  A  C_  P )
81, 5, 7syl2anc 642 1  |-  ( ph  ->  A  C_  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ` cfv 5255  PPointscpoints 26056  PLinescplines 26058  Igcig 26060
This theorem is referenced by:  lineval12a  26084  iscol3  26094  isconcl5ab  26102  isconcl6a  26103  bsstrs  26146
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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