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Theorem usgra1v 21362
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 21324 . . . . 5  |-  ( { A } USGrph  E  ->  ( { A }  e.  _V  /\  E  e.  _V ) )
2 isusgra 21326 . . . . . . . . 9  |-  ( ( { A }  e.  _V  /\  E  e.  _V )  ->  ( { A } USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 } ) )
32adantr 452 . . . . . . . 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 2405 . . . . . . . . . 10  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  E  =  E )
5 eqidd 2405 . . . . . . . . . 10  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  dom  E  =  dom  E )
6 pwsn 3969 . . . . . . . . . . . . . 14  |-  ~P { A }  =  { (/)
,  { A } }
76difeq1i 3421 . . . . . . . . . . . . 13  |-  ( ~P { A }  \  { (/) } )  =  ( { (/) ,  { A } }  \  { (/)
} )
8 snnzg 3881 . . . . . . . . . . . . . . . 16  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
98necomd 2650 . . . . . . . . . . . . . . 15  |-  ( A  e.  _V  ->  (/)  =/=  { A } )
109adantl 453 . . . . . . . . . . . . . 14  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  (/)  =/=  { A } )
11 difprsn1 3895 . . . . . . . . . . . . . 14  |-  ( (/)  =/=  { A }  ->  ( { (/) ,  { A } }  \  { (/) } )  =  { { A } } )
1210, 11syl 16 . . . . . . . . . . . . 13  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( { (/) ,  { A } }  \  { (/) } )  =  { { A } } )
137, 12syl5eq 2448 . . . . . . . . . . . 12  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( ~P { A }  \  { (/) } )  =  { { A } } )
14 biidd 229 . . . . . . . . . . . 12  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  (
( # `  x )  =  2  <->  ( # `  x
)  =  2 ) )
1513, 14rabeqbidv 2911 . . . . . . . . . . 11  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  { x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 }  =  { x  e. 
{ { A } }  |  ( # `  x
)  =  2 } )
16 hashsng 11602 . . . . . . . . . . . . . . 15  |-  ( A  e.  _V  ->  ( # `
 { A }
)  =  1 )
17 1ne2 10143 . . . . . . . . . . . . . . . . 17  |-  1  =/=  2
18 neeq1 2575 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  { A } )  =  1  ->  ( ( # `  { A } )  =/=  2  <->  1  =/=  2 ) )
1917, 18mpbiri 225 . . . . . . . . . . . . . . . 16  |-  ( (
# `  { A } )  =  1  ->  ( # `  { A } )  =/=  2
)
2019neneqd 2583 . . . . . . . . . . . . . . 15  |-  ( (
# `  { A } )  =  1  ->  -.  ( # `  { A } )  =  2 )
2116, 20syl 16 . . . . . . . . . . . . . 14  |-  ( A  e.  _V  ->  -.  ( # `  { A } )  =  2 )
22 2ne0 10039 . . . . . . . . . . . . . . . . . 18  |-  2  =/=  0
2322necomi 2649 . . . . . . . . . . . . . . . . 17  |-  0  =/=  2
2423a1i 11 . . . . . . . . . . . . . . . 16  |-  ( -.  A  e.  _V  ->  0  =/=  2 )
2524neneqd 2583 . . . . . . . . . . . . . . 15  |-  ( -.  A  e.  _V  ->  -.  0  =  2 )
26 snprc 3831 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2726biimpi 187 . . . . . . . . . . . . . . . . . 18  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
2827fveq2d 5691 . . . . . . . . . . . . . . . . 17  |-  ( -.  A  e.  _V  ->  (
# `  { A } )  =  (
# `  (/) ) )
29 hash0 11601 . . . . . . . . . . . . . . . . 17  |-  ( # `  (/) )  =  0
3028, 29syl6eq 2452 . . . . . . . . . . . . . . . 16  |-  ( -.  A  e.  _V  ->  (
# `  { A } )  =  0 )
3130eqeq1d 2412 . . . . . . . . . . . . . . 15  |-  ( -.  A  e.  _V  ->  ( ( # `  { A } )  =  2  <->  0  =  2 ) )
3225, 31mtbird 293 . . . . . . . . . . . . . 14  |-  ( -.  A  e.  _V  ->  -.  ( # `  { A } )  =  2 )
3321, 32pm2.61i 158 . . . . . . . . . . . . 13  |-  -.  ( # `
 { A }
)  =  2
34 snex 4365 . . . . . . . . . . . . . 14  |-  { A }  e.  _V
35 fveq2 5687 . . . . . . . . . . . . . . . 16  |-  ( x  =  { A }  ->  ( # `  x
)  =  ( # `  { A } ) )
3635eqeq1d 2412 . . . . . . . . . . . . . . 15  |-  ( x  =  { A }  ->  ( ( # `  x
)  =  2  <->  ( # `
 { A }
)  =  2 ) )
3736notbid 286 . . . . . . . . . . . . . 14  |-  ( x  =  { A }  ->  ( -.  ( # `  x )  =  2  <->  -.  ( # `  { A } )  =  2 ) )
3834, 37ralsn 3809 . . . . . . . . . . . . 13  |-  ( A. x  e.  { { A } }  -.  ( # `
 x )  =  2  <->  -.  ( # `  { A } )  =  2 )
3933, 38mpbir 201 . . . . . . . . . . . 12  |-  A. x  e.  { { A } }  -.  ( # `  x
)  =  2
40 rabeq0 3609 . . . . . . . . . . . 12  |-  ( { x  e.  { { A } }  |  (
# `  x )  =  2 }  =  (/)  <->  A. x  e.  { { A } }  -.  ( # `
 x )  =  2 )
4139, 40mpbir 201 . . . . . . . . . . 11  |-  { x  e.  { { A } }  |  ( # `  x
)  =  2 }  =  (/)
4215, 41syl6eq 2452 . . . . . . . . . 10  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  { x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 }  =  (/) )
434, 5, 42f1eq123d 5628 . . . . . . . . 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 5598 . . . . . . . . . 10  |-  ( E : dom  E -1-1-> (/)  ->  E : dom  E --> (/) )
45 f00 5587 . . . . . . . . . . 11  |-  ( E : dom  E --> (/)  <->  ( E  =  (/)  /\  dom  E  =  (/) ) )
4645simplbi 447 . . . . . . . . . 10  |-  ( E : dom  E --> (/)  ->  E  =  (/) )
4744, 46syl 16 . . . . . . . . 9  |-  ( E : dom  E -1-1-> (/)  ->  E  =  (/) )
4843, 47syl6bi 220 . . . . . . . 8  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( E : dom  E -1-1-> {
x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 }  ->  E  =  (/) ) )
493, 48sylbid 207 . . . . . . 7  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( { A } USGrph  E  ->  E  =  (/) ) )
5049ex 424 . . . . . 6  |-  ( ( { A }  e.  _V  /\  E  e.  _V )  ->  ( A  e. 
_V  ->  ( { A } USGrph  E  ->  E  =  (/) ) ) )
5150com23 74 . . . . 5  |-  ( ( { A }  e.  _V  /\  E  e.  _V )  ->  ( { A } USGrph  E  ->  ( A  e.  _V  ->  E  =  (/) ) ) )
521, 51mpcom 34 . . . 4  |-  ( { A } USGrph  E  ->  ( A  e.  _V  ->  E  =  (/) ) )
5352com12 29 . . 3  |-  ( A  e.  _V  ->  ( { A } USGrph  E  ->  E  =  (/) ) )
54 usgra0 21343 . . . . 5  |-  ( { A }  e.  _V  ->  { A } USGrph  (/) )
5534, 54ax-mp 8 . . . 4  |-  { A } USGrph 
(/)
56 breq2 4176 . . . 4  |-  ( E  =  (/)  ->  ( { A } USGrph  E  <->  { A } USGrph 
(/) ) )
5755, 56mpbiri 225 . . 3  |-  ( E  =  (/)  ->  { A } USGrph  E )
5853, 57impbid1 195 . 2  |-  ( A  e.  _V  ->  ( { A } USGrph  E  <->  E  =  (/) ) )
59 breq1 4175 . . . 4  |-  ( { A }  =  (/)  ->  ( { A } USGrph  E  <->  (/) USGrph  E ) )
60 usgra0v 21344 . . . 4  |-  ( (/) USGrph  E  <-> 
E  =  (/) )
6159, 60syl6bb 253 . . 3  |-  ( { A }  =  (/)  ->  ( { A } USGrph  E  <-> 
E  =  (/) ) )
6226, 61sylbi 188 . 2  |-  ( -.  A  e.  _V  ->  ( { A } USGrph  E  <->  E  =  (/) ) )
6358, 62pm2.61i 158 1  |-  ( { A } USGrph  E  <->  E  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   {crab 2670   _Vcvv 2916    \ cdif 3277   (/)c0 3588   ~Pcpw 3759   {csn 3774   {cpr 3775   class class class wbr 4172   dom cdm 4837   -->wf 5409   -1-1->wf1 5410   ` cfv 5413   0cc0 8946   1c1 8947   2c2 10005   #chash 11573   USGrph cusg 21318
This theorem is referenced by:  usgrafisindb1  21376  vdfrgra0  28126  vdgfrgra0  28127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-hash 11574  df-usgra 21320
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