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Theorem usgra1v 24511
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 24459 . . . . 5  |-  ( { A } USGrph  E  ->  ( { A }  e.  _V  /\  E  e.  _V ) )
2 isusgra 24465 . . . . . . . . 9  |-  ( ( { A }  e.  _V  /\  E  e.  _V )  ->  ( { A } USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 } ) )
32adantr 463 . . . . . . . 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 2383 . . . . . . . . . 10  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  E  =  E )
5 eqidd 2383 . . . . . . . . . 10  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  dom  E  =  dom  E )
6 pwsn 4157 . . . . . . . . . . . . . 14  |-  ~P { A }  =  { (/)
,  { A } }
76difeq1i 3532 . . . . . . . . . . . . 13  |-  ( ~P { A }  \  { (/) } )  =  ( { (/) ,  { A } }  \  { (/)
} )
8 snnzg 4061 . . . . . . . . . . . . . . . 16  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
98necomd 2653 . . . . . . . . . . . . . . 15  |-  ( A  e.  _V  ->  (/)  =/=  { A } )
109adantl 464 . . . . . . . . . . . . . 14  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  (/)  =/=  { A } )
11 difprsn1 4080 . . . . . . . . . . . . . 14  |-  ( (/)  =/=  { A }  ->  ( { (/) ,  { A } }  \  { (/) } )  =  { { A } } )
1210, 11syl 16 . . . . . . . . . . . . 13  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  ( { (/) ,  { A } }  \  { (/) } )  =  { { A } } )
137, 12syl5eq 2435 . . . . . . . . . . . 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 3029 . . . . . . . . . . 11  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  { x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 }  =  { x  e. 
{ { A } }  |  ( # `  x
)  =  2 } )
16 hashsng 12341 . . . . . . . . . . . . . . 15  |-  ( A  e.  _V  ->  ( # `
 { A }
)  =  1 )
17 1ne2 10665 . . . . . . . . . . . . . . . . 17  |-  1  =/=  2
18 neeq1 2663 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  { A } )  =  1  ->  ( ( # `  { A } )  =/=  2  <->  1  =/=  2 ) )
1917, 18mpbiri 233 . . . . . . . . . . . . . . . 16  |-  ( (
# `  { A } )  =  1  ->  ( # `  { A } )  =/=  2
)
2019neneqd 2584 . . . . . . . . . . . . . . 15  |-  ( (
# `  { A } )  =  1  ->  -.  ( # `  { A } )  =  2 )
2116, 20syl 16 . . . . . . . . . . . . . 14  |-  ( A  e.  _V  ->  -.  ( # `  { A } )  =  2 )
22 0ne2 10664 . . . . . . . . . . . . . . . . 17  |-  0  =/=  2
2322a1i 11 . . . . . . . . . . . . . . . 16  |-  ( -.  A  e.  _V  ->  0  =/=  2 )
2423neneqd 2584 . . . . . . . . . . . . . . 15  |-  ( -.  A  e.  _V  ->  -.  0  =  2 )
25 snprc 4007 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2625biimpi 194 . . . . . . . . . . . . . . . . . 18  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
2726fveq2d 5778 . . . . . . . . . . . . . . . . 17  |-  ( -.  A  e.  _V  ->  (
# `  { A } )  =  (
# `  (/) ) )
28 hash0 12340 . . . . . . . . . . . . . . . . 17  |-  ( # `  (/) )  =  0
2927, 28syl6eq 2439 . . . . . . . . . . . . . . . 16  |-  ( -.  A  e.  _V  ->  (
# `  { A } )  =  0 )
3029eqeq1d 2384 . . . . . . . . . . . . . . 15  |-  ( -.  A  e.  _V  ->  ( ( # `  { A } )  =  2  <->  0  =  2 ) )
3124, 30mtbird 299 . . . . . . . . . . . . . 14  |-  ( -.  A  e.  _V  ->  -.  ( # `  { A } )  =  2 )
3221, 31pm2.61i 164 . . . . . . . . . . . . 13  |-  -.  ( # `
 { A }
)  =  2
33 snex 4603 . . . . . . . . . . . . . 14  |-  { A }  e.  _V
34 fveq2 5774 . . . . . . . . . . . . . . . 16  |-  ( x  =  { A }  ->  ( # `  x
)  =  ( # `  { A } ) )
3534eqeq1d 2384 . . . . . . . . . . . . . . 15  |-  ( x  =  { A }  ->  ( ( # `  x
)  =  2  <->  ( # `
 { A }
)  =  2 ) )
3635notbid 292 . . . . . . . . . . . . . 14  |-  ( x  =  { A }  ->  ( -.  ( # `  x )  =  2  <->  -.  ( # `  { A } )  =  2 ) )
3733, 36ralsn 3983 . . . . . . . . . . . . 13  |-  ( A. x  e.  { { A } }  -.  ( # `
 x )  =  2  <->  -.  ( # `  { A } )  =  2 )
3832, 37mpbir 209 . . . . . . . . . . . 12  |-  A. x  e.  { { A } }  -.  ( # `  x
)  =  2
39 rabeq0 3734 . . . . . . . . . . . 12  |-  ( { x  e.  { { A } }  |  (
# `  x )  =  2 }  =  (/)  <->  A. x  e.  { { A } }  -.  ( # `
 x )  =  2 )
4038, 39mpbir 209 . . . . . . . . . . 11  |-  { x  e.  { { A } }  |  ( # `  x
)  =  2 }  =  (/)
4115, 40syl6eq 2439 . . . . . . . . . 10  |-  ( ( ( { A }  e.  _V  /\  E  e. 
_V )  /\  A  e.  _V )  ->  { x  e.  ( ~P { A }  \  { (/) } )  |  ( # `  x
)  =  2 }  =  (/) )
424, 5, 41f1eq123d 5719 . . . . . . . . 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 5689 . . . . . . . . . 10  |-  ( E : dom  E -1-1-> (/)  ->  E : dom  E --> (/) )
44 f00 5675 . . . . . . . . . . 11  |-  ( E : dom  E --> (/)  <->  ( E  =  (/)  /\  dom  E  =  (/) ) )
4544simplbi 458 . . . . . . . . . 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 432 . . . . . 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 24491 . . . . 5  |-  ( { A }  e.  _V  ->  { A } USGrph  (/) )
5433, 53ax-mp 5 . . . 4  |-  { A } USGrph 
(/)
55 breq2 4371 . . . 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 4370 . . . 4  |-  ( { A }  =  (/)  ->  ( { A } USGrph  E  <->  (/) USGrph  E ) )
59 usgra0v 24492 . . . 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 367    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   {crab 2736   _Vcvv 3034    \ cdif 3386   (/)c0 3711   ~Pcpw 3927   {csn 3944   {cpr 3946   class class class wbr 4367   dom cdm 4913   -->wf 5492   -1-1->wf1 5493   ` cfv 5496   0cc0 9403   1c1 9404   2c2 10502   #chash 12307   USGrph cusg 24451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-hash 12308  df-usgra 24454
This theorem is referenced by:  usgrafisindb1  24530  vdfrgra0  25143  usgo1s0ALT  32755  usgo1s0  32760
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