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Theorem frgrawopreglem1 24836
Description: Lemma 1 for frgrawopreg 24841. In a friendship graph, the classes A and B are sets. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
Hypotheses
Ref Expression
frgrawopreg.a  |-  A  =  { x  e.  V  |  ( ( V VDeg 
E ) `  x
)  =  K }
frgrawopreg.b  |-  B  =  ( V  \  A
)
Assertion
Ref Expression
frgrawopreglem1  |-  ( V FriendGrph  E  ->  ( A  e. 
_V  /\  B  e.  _V ) )
Distinct variable groups:    x, A    x, E    x, K    x, V
Allowed substitution hint:    B( x)

Proof of Theorem frgrawopreglem1
StepHypRef Expression
1 frisusgra 24783 . 2  |-  ( V FriendGrph  E  ->  V USGrph  E )
2 usgrav 24129 . 2  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
3 frgrawopreg.a . . . . 5  |-  A  =  { x  e.  V  |  ( ( V VDeg 
E ) `  x
)  =  K }
4 rabexg 4602 . . . . 5  |-  ( V  e.  _V  ->  { x  e.  V  |  (
( V VDeg  E ) `  x )  =  K }  e.  _V )
53, 4syl5eqel 2559 . . . 4  |-  ( V  e.  _V  ->  A  e.  _V )
6 frgrawopreg.b . . . . 5  |-  B  =  ( V  \  A
)
7 difexg 4600 . . . . 5  |-  ( V  e.  _V  ->  ( V  \  A )  e. 
_V )
86, 7syl5eqel 2559 . . . 4  |-  ( V  e.  _V  ->  B  e.  _V )
95, 8jca 532 . . 3  |-  ( V  e.  _V  ->  ( A  e.  _V  /\  B  e.  _V ) )
109adantr 465 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( A  e.  _V  /\  B  e.  _V )
)
111, 2, 103syl 20 1  |-  ( V FriendGrph  E  ->  ( A  e. 
_V  /\  B  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2821   _Vcvv 3118    \ cdif 3478   class class class wbr 4452   ` cfv 5593  (class class class)co 6294   USGrph cusg 24121   VDeg cvdg 24684   FriendGrph cfrgra 24779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-dm 5014  df-rn 5015  df-usgra 24124  df-frgra 24780
This theorem is referenced by:  frgrawopreglem5  24840  frgrawopreg  24841  frgrawopreg1  24842  frgrawopreg2  24843
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