MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgra1v Structured version   Unicode version

Theorem usgra1v 21440
Description: A class with one (or no) vertex is a simple graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
Assertion
Ref Expression
usgra1v  |-  ( { A } USGrph  E  <->  E  =  (/) )

Proof of Theorem usgra1v
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 usgrav 21402 . . . . 5  |-  ( { A } USGrph  E  ->  ( { A }  e.  _V  /\  E  e.  _V ) )
2 isusgra 21404 . . . . . . . . 9  |-  ( ( { A }  e.  _V  /\  E  e.  _V )  ->  ( { A } USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 } ) )
32adantr 453 . . . . . . . 8  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( { A } USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 } ) )
4 eqidd 2443 . . . . . . . . . 10  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  E  =  E )
5 eqidd 2443 . . . . . . . . . 10  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  dom  E  =  dom  E )
6 pwsn 4033 . . . . . . . . . . . . . 14  |-  ~P { A }  =  { (/)
,  { A } }
76difeq1i 3447 . . . . . . . . . . . . 13  |-  ( ~P { A }  \  { (/) } )  =  ( { (/) ,  { A } }  \  { (/)
} )
8 snnzg 3945 . . . . . . . . . . . . . . . 16  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
98necomd 2693 . . . . . . . . . . . . . . 15  |-  ( A  e.  _V  ->  (/)  =/=  { A } )
109adantl 454 . . . . . . . . . . . . . 14  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  (/)  =/=  { A } )
11 difprsn1 3959 . . . . . . . . . . . . . 14  |-  ( (/)  =/=  { A }  ->  ( { (/) ,  { A } }  \  { (/) } )  =  { { A } } )
1210, 11syl 16 . . . . . . . . . . . . 13  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( { (/) ,  { A } }  \  { (/) } )  =  { { A } } )
137, 12syl5eq 2486 . . . . . . . . . . . 12  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( ~P { A }  \  { (/) } )  =  { { A } } )
14 biidd 230 . . . . . . . . . . . 12  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  (
( # `  x )  =  2  <->  ( # `  x
)  =  2 ) )
1513, 14rabeqbidv 2957 . . . . . . . . . . 11  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  { x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 }  =  { x  e. 
{ { A } }  |  ( # `  x
)  =  2 } )
16 hashsng 11678 . . . . . . . . . . . . . . 15  |-  ( A  e.  _V  ->  ( # `
 { A }
)  =  1 )
17 1ne2 10218 . . . . . . . . . . . . . . . . 17  |-  1  =/=  2
18 neeq1 2615 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  { A } )  =  1  ->  ( ( # `  { A } )  =/=  2  <->  1  =/=  2 ) )
1917, 18mpbiri 226 . . . . . . . . . . . . . . . 16  |-  ( (
# `  { A } )  =  1  ->  ( # `  { A } )  =/=  2
)
2019neneqd 2623 . . . . . . . . . . . . . . 15  |-  ( (
# `  { A } )  =  1  ->  -.  ( # `  { A } )  =  2 )
2116, 20syl 16 . . . . . . . . . . . . . 14  |-  ( A  e.  _V  ->  -.  ( # `  { A } )  =  2 )
22 2ne0 10114 . . . . . . . . . . . . . . . . . 18  |-  2  =/=  0
2322necomi 2692 . . . . . . . . . . . . . . . . 17  |-  0  =/=  2
2423a1i 11 . . . . . . . . . . . . . . . 16  |-  ( -.  A  e.  _V  ->  0  =/=  2 )
2524neneqd 2623 . . . . . . . . . . . . . . 15  |-  ( -.  A  e.  _V  ->  -.  0  =  2 )
26 snprc 3895 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2726biimpi 188 . . . . . . . . . . . . . . . . . 18  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
2827fveq2d 5761 . . . . . . . . . . . . . . . . 17  |-  ( -.  A  e.  _V  ->  (
# `  { A } )  =  (
# `  (/) ) )
29 hash0 11677 . . . . . . . . . . . . . . . . 17  |-  ( # `  (/) )  =  0
3028, 29syl6eq 2490 . . . . . . . . . . . . . . . 16  |-  ( -.  A  e.  _V  ->  (
# `  { A } )  =  0 )
3130eqeq1d 2450 . . . . . . . . . . . . . . 15  |-  ( -.  A  e.  _V  ->  ( ( # `  { A } )  =  2  <->  0  =  2 ) )
3225, 31mtbird 294 . . . . . . . . . . . . . 14  |-  ( -.  A  e.  _V  ->  -.  ( # `  { A } )  =  2 )
3321, 32pm2.61i 159 . . . . . . . . . . . . 13  |-  -.  ( # `
 { A }
)  =  2
34 snex 4434 . . . . . . . . . . . . . 14  |-  { A }  e.  _V
35 fveq2 5757 . . . . . . . . . . . . . . . 16  |-  ( x  =  { A }  ->  ( # `  x
)  =  ( # `  { A } ) )
3635eqeq1d 2450 . . . . . . . . . . . . . . 15  |-  ( x  =  { A }  ->  ( ( # `  x
)  =  2  <->  ( # `
 { A }
)  =  2 ) )
3736notbid 287 . . . . . . . . . . . . . 14  |-  ( x  =  { A }  ->  ( -.  ( # `  x )  =  2  <->  -.  ( # `  { A } )  =  2 ) )
3834, 37ralsn 3873 . . . . . . . . . . . . 13  |-  ( A. x  e.  { { A } }  -.  ( # `
 x )  =  2  <->  -.  ( # `  { A } )  =  2 )
3933, 38mpbir 202 . . . . . . . . . . . 12  |-  A. x  e.  { { A } }  -.  ( # `  x
)  =  2
40 rabeq0 3634 . . . . . . . . . . . 12  |-  ( { x  e.  { { A } }  |  (
# `  x )  =  2 }  =  (/)  <->  A. x  e.  { { A } }  -.  ( # `
 x )  =  2 )
4139, 40mpbir 202 . . . . . . . . . . 11  |-  { x  e.  { { A } }  |  ( # `  x
)  =  2 }  =  (/)
4215, 41syl6eq 2490 . . . . . . . . . 10  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  { x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 }  =  (/) )
434, 5, 42f1eq123d 5698 . . . . . . . . 9  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( E : dom  E -1-1-> {
x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 }  <-> 
E : dom  E -1-1-> (/) ) )
44 f1f 5668 . . . . . . . . . 10  |-  ( E : dom  E -1-1-> (/)  ->  E : dom  E --> (/) )
45 f00 5657 . . . . . . . . . . 11  |-  ( E : dom  E --> (/)  <->  ( E  =  (/)  /\  dom  E  =  (/) ) )
4645simplbi 448 . . . . . . . . . 10  |-  ( E : dom  E --> (/)  ->  E  =  (/) )
4744, 46syl 16 . . . . . . . . 9  |-  ( E : dom  E -1-1-> (/)  ->  E  =  (/) )
4843, 47syl6bi 221 . . . . . . . 8  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( E : dom  E -1-1-> {
x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 }  ->  E  =  (/) ) )
493, 48sylbid 208 . . . . . . 7  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( { A } USGrph  E  ->  E  =  (/) ) )
5049ex 425 . . . . . 6  |-  ( ( { A }  e.  _V  /\  E  e.  _V )  ->  ( A  e. 
_V  ->  ( { A } USGrph  E  ->  E  =  (/) ) ) )
5150com23 75 . . . . 5  |-  ( ( { A }  e.  _V  /\  E  e.  _V )  ->  ( { A } USGrph  E  ->  ( A  e.  _V  ->  E  =  (/) ) ) )
521, 51mpcom 35 . . . 4  |-  ( { A } USGrph  E  ->  ( A  e.  _V  ->  E  =  (/) ) )
5352com12 30 . . 3  |-  ( A  e.  _V  ->  ( { A } USGrph  E  ->  E  =  (/) ) )
54 usgra0 21421 . . . . 5  |-  ( { A }  e.  _V  ->  { A } USGrph  (/) )
5534, 54ax-mp 5 . . . 4  |-  { A } USGrph 
(/)
56 breq2 4241 . . . 4  |-  ( E  =  (/)  ->  ( { A } USGrph  E  <->  { A } USGrph 
(/) ) )
5755, 56mpbiri 226 . . 3  |-  ( E  =  (/)  ->  { A } USGrph  E )
5853, 57impbid1 196 . 2  |-  ( A  e.  _V  ->  ( { A } USGrph  E  <->  E  =  (/) ) )
59 breq1 4240 . . . 4  |-  ( { A }  =  (/)  ->  ( { A } USGrph  E  <->  (/) USGrph  E ) )
60 usgra0v 21422 . . . 4  |-  ( (/) USGrph  E  <-> 
E  =  (/) )
6159, 60syl6bb 254 . . 3  |-  ( { A }  =  (/)  ->  ( { A } USGrph  E  <-> 
E  =  (/) ) )
6226, 61sylbi 189 . 2  |-  ( -.  A  e.  _V  ->  ( { A } USGrph  E  <->  E  =  (/) ) )
6358, 62pm2.61i 159 1  |-  ( { A } USGrph  E  <->  E  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605   A.wral 2711   {crab 2715   _Vcvv 2962    \ cdif 3303   (/)c0 3613   ~Pcpw 3823   {csn 3838   {cpr 3839   class class class wbr 4237   dom cdm 4907   -->wf 5479   -1-1->wf1 5480   ` cfv 5483   0cc0 9021   1c1 9022   2c2 10080   #chash 11649   USGrph cusg 21396
This theorem is referenced by:  usgrafisindb1  21454  vdfrgra0  28510  vdgfrgra0  28511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-card 7857  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-n0 10253  df-z 10314  df-uz 10520  df-fz 11075  df-hash 11650  df-usgra 21398
  Copyright terms: Public domain W3C validator