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Theorem frgrawopreglem1 25617
Description: Lemma 1 for frgrawopreg 25622. In a friendship graph, the classes A and B are sets. The definition of A and B corresponds to definition 3 in [Huneke] p. 2: "Let A be the set of all vertices of degree k, let B be the set of all vertices of degree different from k, ..." (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 25565 . 2  |-  ( V FriendGrph  E  ->  V USGrph  E )
2 usgrav 24911 . 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 4575 . . . . 5  |-  ( V  e.  _V  ->  { x  e.  V  |  (
( V VDeg  E ) `  x )  =  K }  e.  _V )
53, 4syl5eqel 2521 . . . 4  |-  ( V  e.  _V  ->  A  e.  _V )
6 frgrawopreg.b . . . . 5  |-  B  =  ( V  \  A
)
7 difexg 4573 . . . . 5  |-  ( V  e.  _V  ->  ( V  \  A )  e. 
_V )
86, 7syl5eqel 2521 . . . 4  |-  ( V  e.  _V  ->  B  e.  _V )
95, 8jca 534 . . 3  |-  ( V  e.  _V  ->  ( A  e.  _V  /\  B  e.  _V ) )
109adantr 466 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( A  e.  _V  /\  B  e.  _V )
)
111, 2, 103syl 18 1  |-  ( V FriendGrph  E  ->  ( A  e. 
_V  /\  B  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   {crab 2786   _Vcvv 3087    \ cdif 3439   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   USGrph cusg 24903   VDeg cvdg 25466   FriendGrph cfrgra 25561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-rel 4861  df-cnv 4862  df-dm 4864  df-rn 4865  df-usgra 24906  df-frgra 25562
This theorem is referenced by:  frgrawopreglem5  25621  frgrawopreg  25622  frgrawopreg1  25623  frgrawopreg2  25624
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