Theorem usgrprc 26158

Description: The class of simple graphs is a proper class (and therefore, because of prcssprc 4806, the classes of multigraphs, pseudographs and hypergraphs are proper classes, too). (Contributed by AV, 27-Dec-2020.)

Assertion

Ref	Expression
usgrprc	|- USGraph e/ _V

Proof of Theorem usgrprc

Dummy variables v e are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- { <. v , e >. | e : (/) --> (/) } = { <. v , e >. | e : (/) --> (/) }
2	1	griedg0ssusgr 26157	. 2 |- { <. v , e >. | e : (/) --> (/) } C_ USGraph
3	1	griedg0prc 26156	. 2 |- { <. v , e >. | e : (/) --> (/) } e/ _V
4		prcssprc 4806	. 2 |- ( ( { <. v , e >. | e : (/) --> (/) } C_ USGraph /\ { <. v , e >. | e : (/) --> (/) } e/ _V ) -> USGraph e/ _V )
5	2, 3, 4	mp2an 708	1 |- USGraph e/ _V

Syntax hints:   e/ wnel 2897   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {copab 4712   -->wf 5884   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-nel 2898  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: (None)