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Theorem isconcl3b 26099
Description: Only lines can be concurrent. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypotheses
Ref Expression
isconclb  |-  ( ph  ->  F  e. Ig )
isconcl3.1  |-  ( ph  ->  A  e.  (con `  F ) )
Assertion
Ref Expression
isconcl3b  |-  ( ph  ->  A  C_  (PLines `  F
) )

Proof of Theorem isconcl3b
StepHypRef Expression
1 isconcl3.1 . . . 4  |-  ( ph  ->  A  e.  (con `  F ) )
2 isconclb . . . . 5  |-  ( ph  ->  F  e. Ig )
32isconcl2b 26098 . . . 4  |-  ( ph  ->  ( A  e.  (con
`  F )  <->  ( A  e.  ~P (PLines `  F
)  /\  |^| A  =/=  (/) ) ) )
41, 3mpbid 201 . . 3  |-  ( ph  ->  ( A  e.  ~P (PLines `  F )  /\  |^| A  =/=  (/) ) )
54simpld 445 . 2  |-  ( ph  ->  A  e.  ~P (PLines `  F ) )
6 elpwi 3633 . 2  |-  ( A  e.  ~P (PLines `  F )  ->  A  C_  (PLines `  F )
)
75, 6syl 15 1  |-  ( ph  ->  A  C_  (PLines `  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684    =/= wne 2446    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   |^|cint 3862   ` cfv 5255  PLinescplines 26058  Igcig 26060  conccon2 26095
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
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-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  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-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-con2 26096
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