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Theorem isconcl2b 26201
Description: The predicate "are concurrent lines". (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypothesis
Ref Expression
isconclb  |-  ( ph  ->  F  e. Ig )
Assertion
Ref Expression
isconcl2b  |-  ( ph  ->  ( A  e.  (con
`  F )  <->  ( A  e.  ~P (PLines `  F
)  /\  |^| A  =/=  (/) ) ) )

Proof of Theorem isconcl2b
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isconclb . . . 4  |-  ( ph  ->  F  e. Ig )
21isconcl1b 26200 . . 3  |-  ( ph  ->  (con `  F )  =  { x  e.  ~P (PLines `  F )  | 
|^| x  =/=  (/) } )
32eleq2d 2363 . 2  |-  ( ph  ->  ( A  e.  (con
`  F )  <->  A  e.  { x  e.  ~P (PLines `  F )  |  |^| x  =/=  (/) } ) )
4 inteq 3881 . . . 4  |-  ( x  =  A  ->  |^| x  =  |^| A )
54neeq1d 2472 . . 3  |-  ( x  =  A  ->  ( |^| x  =/=  (/)  <->  |^| A  =/=  (/) ) )
65elrab 2936 . 2  |-  ( A  e.  { x  e. 
~P (PLines `  F )  |  |^| x  =/=  (/) }  <->  ( A  e.  ~P (PLines `  F
)  /\  |^| A  =/=  (/) ) )
73, 6syl6bb 252 1  |-  ( ph  ->  ( A  e.  (con
`  F )  <->  ( A  e.  ~P (PLines `  F
)  /\  |^| A  =/=  (/) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   (/)c0 3468   ~Pcpw 3638   |^|cint 3878   ` cfv 5271  PLinescplines 26161  Igcig 26163  conccon2 26198
This theorem is referenced by:  isconcl3b  26202  isconcl4b  26203
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  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-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-con2 26199
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