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.

Step	Hyp	Ref	Expression
1		griedg0prc.u	. . . . 5 |- U = { <. v , e >. | e : (/) --> (/) }
2	1	eleq2i 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 : (/) --> (/) ) )
4	2, 3	bitri 264	. . 3 |- ( g e. U <-> E. v E. e ( g = <. v , e >. /\ e : (/) --> (/) ) )
5		opex 4932	. . . . . . . 8 |- <. v , e >. e. _V
6	5	a1i 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 )
10	7, 8, 9	mp2an 708	. . . . . . . 8 |- (iEdg ` <. v , e >. ) = e
11		f0bi 6088	. . . . . . . . 9 |- ( e : (/) --> (/) <-> e = (/) )
12	11	biimpi 206	. . . . . . . 8 |- ( e : (/) --> (/) -> e = (/) )
13	10, 12	syl5eq 2668	. . . . . . 7 |- ( e : (/) --> (/) -> (iEdg ` <. v , e >. ) = (/) )
14	6, 13	usgr0e 26128	. . . . . 6 |- ( e : (/) --> (/) -> <. v , e >. e. USGraph )
15	14	adantl 482	. . . . 5 |- ( ( g = <. v , e >. /\ e : (/) --> (/) ) -> <. v , e >. e. USGraph )
16		eleq1 2689	. . . . . 6 |- ( g = <. v , e >. -> ( g e. USGraph <-> <. v , e >. e. USGraph ) )
17	16	adantr 481	. . . . 5 |- ( ( g = <. v , e >. /\ e : (/) --> (/) ) -> ( g e. USGraph <-> <. v , e >. e. USGraph ) )
18	15, 17	mpbird 247	. . . 4 |- ( ( g = <. v , e >. /\ e : (/) --> (/) ) -> g e. USGraph )
19	18	exlimivv 1860	. . 3 |- ( E. v E. e ( g = <. v , e >. /\ e : (/) --> (/) ) -> g e. USGraph )
20	4, 19	sylbi 207	. 2 |- ( g e. U -> g e. USGraph )
21	20	ssriv 3607	1 |- U C_ USGraph

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