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Theorem usgra1v 21401
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 21363 . . . . 5  |-  ( { A } USGrph  E  ->  ( { A }  e.  _V  /\  E  e.  _V ) )
2 isusgra 21365 . . . . . . . . 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 2436 . . . . . . . . . 10  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  E  =  E )
5 eqidd 2436 . . . . . . . . . 10  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  dom  E  =  dom  E )
6 pwsn 4001 . . . . . . . . . . . . . 14  |-  ~P { A }  =  { (/)
,  { A } }
76difeq1i 3453 . . . . . . . . . . . . 13  |-  ( ~P { A }  \  { (/) } )  =  ( { (/) ,  { A } }  \  { (/)
} )
8 snnzg 3913 . . . . . . . . . . . . . . . 16  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
98necomd 2681 . . . . . . . . . . . . . . 15  |-  ( A  e.  _V  ->  (/)  =/=  { A } )
109adantl 453 . . . . . . . . . . . . . 14  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  (/)  =/=  { A } )
11 difprsn1 3927 . . . . . . . . . . . . . 14  |-  ( (/)  =/=  { A }  ->  ( { (/) ,  { A } }  \  { (/) } )  =  { { A } } )
1210, 11syl 16 . . . . . . . . . . . . 13  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( { (/) ,  { A } }  \  { (/) } )  =  { { A } } )
137, 12syl5eq 2479 . . . . . . . . . . . 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 2943 . . . . . . . . . . 11  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  { x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 }  =  { x  e. 
{ { A } }  |  ( # `  x
)  =  2 } )
16 hashsng 11639 . . . . . . . . . . . . . . 15  |-  ( A  e.  _V  ->  ( # `
 { A }
)  =  1 )
17 1ne2 10179 . . . . . . . . . . . . . . . . 17  |-  1  =/=  2
18 neeq1 2606 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  { A } )  =  1  ->  ( ( # `  { A } )  =/=  2  <->  1  =/=  2 ) )
1917, 18mpbiri 225 . . . . . . . . . . . . . . . 16  |-  ( (
# `  { A } )  =  1  ->  ( # `  { A } )  =/=  2
)
2019neneqd 2614 . . . . . . . . . . . . . . 15  |-  ( (
# `  { A } )  =  1  ->  -.  ( # `  { A } )  =  2 )
2116, 20syl 16 . . . . . . . . . . . . . 14  |-  ( A  e.  _V  ->  -.  ( # `  { A } )  =  2 )
22 2ne0 10075 . . . . . . . . . . . . . . . . . 18  |-  2  =/=  0
2322necomi 2680 . . . . . . . . . . . . . . . . 17  |-  0  =/=  2
2423a1i 11 . . . . . . . . . . . . . . . 16  |-  ( -.  A  e.  _V  ->  0  =/=  2 )
2524neneqd 2614 . . . . . . . . . . . . . . 15  |-  ( -.  A  e.  _V  ->  -.  0  =  2 )
26 snprc 3863 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2726biimpi 187 . . . . . . . . . . . . . . . . . 18  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
2827fveq2d 5724 . . . . . . . . . . . . . . . . 17  |-  ( -.  A  e.  _V  ->  (
# `  { A } )  =  (
# `  (/) ) )
29 hash0 11638 . . . . . . . . . . . . . . . . 17  |-  ( # `  (/) )  =  0
3028, 29syl6eq 2483 . . . . . . . . . . . . . . . 16  |-  ( -.  A  e.  _V  ->  (
# `  { A } )  =  0 )
3130eqeq1d 2443 . . . . . . . . . . . . . . 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 4397 . . . . . . . . . . . . . 14  |-  { A }  e.  _V
35 fveq2 5720 . . . . . . . . . . . . . . . 16  |-  ( x  =  { A }  ->  ( # `  x
)  =  ( # `  { A } ) )
3635eqeq1d 2443 . . . . . . . . . . . . . . 15  |-  ( x  =  { A }  ->  ( ( # `  x
)  =  2  <->  ( # `
 { A }
)  =  2 ) )
3736notbid 286 . . . . . . . . . . . . . 14  |-  ( x  =  { A }  ->  ( -.  ( # `  x )  =  2  <->  -.  ( # `  { A } )  =  2 ) )
3834, 37ralsn 3841 . . . . . . . . . . . . 13  |-  ( A. x  e.  { { A } }  -.  ( # `
 x )  =  2  <->  -.  ( # `  { A } )  =  2 )
3933, 38mpbir 201 . . . . . . . . . . . 12  |-  A. x  e.  { { A } }  -.  ( # `  x
)  =  2
40 rabeq0 3641 . . . . . . . . . . . 12  |-  ( { x  e.  { { A } }  |  (
# `  x )  =  2 }  =  (/)  <->  A. x  e.  { { A } }  -.  ( # `
 x )  =  2 )
4139, 40mpbir 201 . . . . . . . . . . 11  |-  { x  e.  { { A } }  |  ( # `  x
)  =  2 }  =  (/)
4215, 41syl6eq 2483 . . . . . . . . . 10  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  { x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 }  =  (/) )
434, 5, 42f1eq123d 5661 . . . . . . . . 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 5631 . . . . . . . . . 10  |-  ( E : dom  E -1-1-> (/)  ->  E : dom  E --> (/) )
45 f00 5620 . . . . . . . . . . 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 21382 . . . . 5  |-  ( { A }  e.  _V  ->  { A } USGrph  (/) )
5534, 54ax-mp 8 . . . 4  |-  { A } USGrph 
(/)
56 breq2 4208 . . . 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 4207 . . . 4  |-  ( { A }  =  (/)  ->  ( { A } USGrph  E  <->  (/) USGrph  E ) )
60 usgra0v 21383 . . . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   {crab 2701   _Vcvv 2948    \ cdif 3309   (/)c0 3620   ~Pcpw 3791   {csn 3806   {cpr 3807   class class class wbr 4204   dom cdm 4870   -->wf 5442   -1-1->wf1 5443   ` cfv 5446   0cc0 8982   1c1 8983   2c2 10041   #chash 11610   USGrph cusg 21357
This theorem is referenced by:  usgrafisindb1  21415  vdfrgra0  28349  vdgfrgra0  28350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-hash 11611  df-usgra 21359
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