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Theorem griedg0ssusgr 26157
Description: The class of all simple graphs is a superclass of the class of empty graphs represented as ordered pairs. (Contributed by AV, 27-Dec-2020.)
Hypothesis
Ref Expression
griedg0prc.u  |-  U  =  { <. v ,  e
>.  |  e : (/) --> (/)
}
Assertion
Ref Expression
griedg0ssusgr  |-  U  C_ USGraph
Distinct variable group:    v, e
Allowed substitution hints:    U( v, e)

Proof of Theorem griedg0ssusgr
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 griedg0prc.u . . . . 5  |-  U  =  { <. v ,  e
>.  |  e : (/) --> (/)
}
21eleq2i 2693 . . . 4  |-  ( g  e.  U  <->  g  e.  {
<. v ,  e >.  |  e : (/) --> (/) } )
3 elopab 4983 . . . 4  |-  ( g  e.  { <. v ,  e >.  |  e : (/) --> (/) }  <->  E. v E. e ( g  = 
<. v ,  e >.  /\  e : (/) --> (/) ) )
42, 3bitri 264 . . 3  |-  ( g  e.  U  <->  E. v E. e ( g  = 
<. v ,  e >.  /\  e : (/) --> (/) ) )
5 opex 4932 . . . . . . . 8  |-  <. v ,  e >.  e.  _V
65a1i 11 . . . . . . 7  |-  ( e : (/) --> (/)  ->  <. v ,  e >.  e.  _V )
7 vex 3203 . . . . . . . . 9  |-  v  e. 
_V
8 vex 3203 . . . . . . . . 9  |-  e  e. 
_V
9 opiedgfv 25887 . . . . . . . . 9  |-  ( ( v  e.  _V  /\  e  e.  _V )  ->  (iEdg `  <. v ,  e >. )  =  e )
107, 8, 9mp2an 708 . . . . . . . 8  |-  (iEdg `  <. v ,  e >.
)  =  e
11 f0bi 6088 . . . . . . . . 9  |-  ( e : (/) --> (/)  <->  e  =  (/) )
1211biimpi 206 . . . . . . . 8  |-  ( e : (/) --> (/)  ->  e  =  (/) )
1310, 12syl5eq 2668 . . . . . . 7  |-  ( e : (/) --> (/)  ->  (iEdg `  <. v ,  e >. )  =  (/) )
146, 13usgr0e 26128 . . . . . 6  |-  ( e : (/) --> (/)  ->  <. v ,  e >.  e. USGraph  )
1514adantl 482 . . . . 5  |-  ( ( g  =  <. v ,  e >.  /\  e : (/) --> (/) )  ->  <. v ,  e >.  e. USGraph  )
16 eleq1 2689 . . . . . 6  |-  ( g  =  <. v ,  e
>.  ->  ( g  e. USGraph  <->  <.
v ,  e >.  e. USGraph  ) )
1716adantr 481 . . . . 5  |-  ( ( g  =  <. v ,  e >.  /\  e : (/) --> (/) )  ->  (
g  e. USGraph  <->  <. v ,  e
>.  e. USGraph  ) )
1815, 17mpbird 247 . . . 4  |-  ( ( g  =  <. v ,  e >.  /\  e : (/) --> (/) )  ->  g  e. USGraph  )
1918exlimivv 1860 . . 3  |-  ( E. v E. e ( g  =  <. v ,  e >.  /\  e : (/) --> (/) )  ->  g  e. USGraph  )
204, 19sylbi 207 . 2  |-  ( g  e.  U  ->  g  e. USGraph  )
2120ssriv 3607 1  |-  U  C_ USGraph
Colors of variables: wff setvar class
Syntax hints:    <-> wb 196    /\ wa 384    = wceq 1483   E.wex 1704    e. wcel 1990   _Vcvv 3200    C_ wss 3574   (/)c0 3915   <.cop 4183   {copab 4712   -->wf 5884   ` cfv 5888  iEdgciedg 25875   USGraph cusgr 26044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896  df-2nd 7169  df-iedg 25877  df-usgr 26046
This theorem is referenced by:  usgrprc  26158
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