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Theorem usgra1v 24262
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 24210 . . . . 5  |-  ( { A } USGrph  E  ->  ( { A }  e.  _V  /\  E  e.  _V ) )
2 isusgra 24216 . . . . . . . . 9  |-  ( ( { A }  e.  _V  /\  E  e.  _V )  ->  ( { A } USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 } ) )
32adantr 465 . . . . . . . 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 2444 . . . . . . . . . 10  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  E  =  E )
5 eqidd 2444 . . . . . . . . . 10  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  dom  E  =  dom  E )
6 pwsn 4228 . . . . . . . . . . . . . 14  |-  ~P { A }  =  { (/)
,  { A } }
76difeq1i 3603 . . . . . . . . . . . . 13  |-  ( ~P { A }  \  { (/) } )  =  ( { (/) ,  { A } }  \  { (/)
} )
8 snnzg 4132 . . . . . . . . . . . . . . . 16  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
98necomd 2714 . . . . . . . . . . . . . . 15  |-  ( A  e.  _V  ->  (/)  =/=  { A } )
109adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  (/)  =/=  { A } )
11 difprsn1 4151 . . . . . . . . . . . . . 14  |-  ( (/)  =/=  { A }  ->  ( { (/) ,  { A } }  \  { (/) } )  =  { { A } } )
1210, 11syl 16 . . . . . . . . . . . . 13  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( { (/) ,  { A } }  \  { (/) } )  =  { { A } } )
137, 12syl5eq 2496 . . . . . . . . . . . 12  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( ~P { A }  \  { (/) } )  =  { { A } } )
14 biidd 237 . . . . . . . . . . . 12  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  (
( # `  x )  =  2  <->  ( # `  x
)  =  2 ) )
1513, 14rabeqbidv 3090 . . . . . . . . . . 11  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  { x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 }  =  { x  e. 
{ { A } }  |  ( # `  x
)  =  2 } )
16 hashsng 12417 . . . . . . . . . . . . . . 15  |-  ( A  e.  _V  ->  ( # `
 { A }
)  =  1 )
17 1ne2 10754 . . . . . . . . . . . . . . . . 17  |-  1  =/=  2
18 neeq1 2724 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  { A } )  =  1  ->  ( ( # `  { A } )  =/=  2  <->  1  =/=  2 ) )
1917, 18mpbiri 233 . . . . . . . . . . . . . . . 16  |-  ( (
# `  { A } )  =  1  ->  ( # `  { A } )  =/=  2
)
2019neneqd 2645 . . . . . . . . . . . . . . 15  |-  ( (
# `  { A } )  =  1  ->  -.  ( # `  { A } )  =  2 )
2116, 20syl 16 . . . . . . . . . . . . . 14  |-  ( A  e.  _V  ->  -.  ( # `  { A } )  =  2 )
22 0ne2 10753 . . . . . . . . . . . . . . . . 17  |-  0  =/=  2
2322a1i 11 . . . . . . . . . . . . . . . 16  |-  ( -.  A  e.  _V  ->  0  =/=  2 )
2423neneqd 2645 . . . . . . . . . . . . . . 15  |-  ( -.  A  e.  _V  ->  -.  0  =  2 )
25 snprc 4078 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2625biimpi 194 . . . . . . . . . . . . . . . . . 18  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
2726fveq2d 5860 . . . . . . . . . . . . . . . . 17  |-  ( -.  A  e.  _V  ->  (
# `  { A } )  =  (
# `  (/) ) )
28 hash0 12416 . . . . . . . . . . . . . . . . 17  |-  ( # `  (/) )  =  0
2927, 28syl6eq 2500 . . . . . . . . . . . . . . . 16  |-  ( -.  A  e.  _V  ->  (
# `  { A } )  =  0 )
3029eqeq1d 2445 . . . . . . . . . . . . . . 15  |-  ( -.  A  e.  _V  ->  ( ( # `  { A } )  =  2  <->  0  =  2 ) )
3124, 30mtbird 301 . . . . . . . . . . . . . 14  |-  ( -.  A  e.  _V  ->  -.  ( # `  { A } )  =  2 )
3221, 31pm2.61i 164 . . . . . . . . . . . . 13  |-  -.  ( # `
 { A }
)  =  2
33 snex 4678 . . . . . . . . . . . . . 14  |-  { A }  e.  _V
34 fveq2 5856 . . . . . . . . . . . . . . . 16  |-  ( x  =  { A }  ->  ( # `  x
)  =  ( # `  { A } ) )
3534eqeq1d 2445 . . . . . . . . . . . . . . 15  |-  ( x  =  { A }  ->  ( ( # `  x
)  =  2  <->  ( # `
 { A }
)  =  2 ) )
3635notbid 294 . . . . . . . . . . . . . 14  |-  ( x  =  { A }  ->  ( -.  ( # `  x )  =  2  <->  -.  ( # `  { A } )  =  2 ) )
3733, 36ralsn 4053 . . . . . . . . . . . . 13  |-  ( A. x  e.  { { A } }  -.  ( # `
 x )  =  2  <->  -.  ( # `  { A } )  =  2 )
3832, 37mpbir 209 . . . . . . . . . . . 12  |-  A. x  e.  { { A } }  -.  ( # `  x
)  =  2
39 rabeq0 3793 . . . . . . . . . . . 12  |-  ( { x  e.  { { A } }  |  (
# `  x )  =  2 }  =  (/)  <->  A. x  e.  { { A } }  -.  ( # `
 x )  =  2 )
4038, 39mpbir 209 . . . . . . . . . . 11  |-  { x  e.  { { A } }  |  ( # `  x
)  =  2 }  =  (/)
4115, 40syl6eq 2500 . . . . . . . . . 10  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  { x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 }  =  (/) )
424, 5, 41f1eq123d 5801 . . . . . . . . 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-> (/) ) )
43 f1f 5771 . . . . . . . . . 10  |-  ( E : dom  E -1-1-> (/)  ->  E : dom  E --> (/) )
44 f00 5757 . . . . . . . . . . 11  |-  ( E : dom  E --> (/)  <->  ( E  =  (/)  /\  dom  E  =  (/) ) )
4544simplbi 460 . . . . . . . . . 10  |-  ( E : dom  E --> (/)  ->  E  =  (/) )
4643, 45syl 16 . . . . . . . . 9  |-  ( E : dom  E -1-1-> (/)  ->  E  =  (/) )
4742, 46syl6bi 228 . . . . . . . 8  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( E : dom  E -1-1-> {
x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 }  ->  E  =  (/) ) )
483, 47sylbid 215 . . . . . . 7  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( { A } USGrph  E  ->  E  =  (/) ) )
4948ex 434 . . . . . 6  |-  ( ( { A }  e.  _V  /\  E  e.  _V )  ->  ( A  e. 
_V  ->  ( { A } USGrph  E  ->  E  =  (/) ) ) )
5049com23 78 . . . . 5  |-  ( ( { A }  e.  _V  /\  E  e.  _V )  ->  ( { A } USGrph  E  ->  ( A  e.  _V  ->  E  =  (/) ) ) )
511, 50mpcom 36 . . . 4  |-  ( { A } USGrph  E  ->  ( A  e.  _V  ->  E  =  (/) ) )
5251com12 31 . . 3  |-  ( A  e.  _V  ->  ( { A } USGrph  E  ->  E  =  (/) ) )
53 usgra0 24242 . . . . 5  |-  ( { A }  e.  _V  ->  { A } USGrph  (/) )
5433, 53ax-mp 5 . . . 4  |-  { A } USGrph 
(/)
55 breq2 4441 . . . 4  |-  ( E  =  (/)  ->  ( { A } USGrph  E  <->  { A } USGrph 
(/) ) )
5654, 55mpbiri 233 . . 3  |-  ( E  =  (/)  ->  { A } USGrph  E )
5752, 56impbid1 203 . 2  |-  ( A  e.  _V  ->  ( { A } USGrph  E  <->  E  =  (/) ) )
58 breq1 4440 . . . 4  |-  ( { A }  =  (/)  ->  ( { A } USGrph  E  <->  (/) USGrph  E ) )
59 usgra0v 24243 . . . 4  |-  ( (/) USGrph  E  <-> 
E  =  (/) )
6058, 59syl6bb 261 . . 3  |-  ( { A }  =  (/)  ->  ( { A } USGrph  E  <-> 
E  =  (/) ) )
6125, 60sylbi 195 . 2  |-  ( -.  A  e.  _V  ->  ( { A } USGrph  E  <->  E  =  (/) ) )
6257, 61pm2.61i 164 1  |-  ( { A } USGrph  E  <->  E  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   {crab 2797   _Vcvv 3095    \ cdif 3458   (/)c0 3770   ~Pcpw 3997   {csn 4014   {cpr 4016   class class class wbr 4437   dom cdm 4989   -->wf 5574   -1-1->wf1 5575   ` cfv 5578   0cc0 9495   1c1 9496   2c2 10591   #chash 12384   USGrph cusg 24202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-n0 10802  df-z 10871  df-uz 11091  df-fz 11682  df-hash 12385  df-usgra 24205
This theorem is referenced by:  usgrafisindb1  24281  vdfrgra0  24894  usgo1s0ALT  32275  usgo1s0  32280
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