Theorem uspgrloopvtxel 26412

Description: A vertex in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 26141). (Contributed by AV, 17-Dec-2020.)

Hypothesis

Ref	Expression
uspgrloopvtx.g	|- G = <. V , { <. A , { N } >. } >.

Assertion

Ref	Expression
uspgrloopvtxel	|- ( ( V e. W /\ N e. V ) -> N e. (Vtx ` G ) )

Proof of Theorem uspgrloopvtxel

Step	Hyp	Ref	Expression
1		uspgrloopvtx.g	. . 3 |- G = <. V , { <. A , { N } >. } >.
2	1	uspgrloopvtx 26411	. 2 |- ( V e. W -> (Vtx ` G ) = V )
3		eleq2 2690	. . . . 5 |- ( V = (Vtx ` G ) -> ( N e. V <-> N e. (Vtx ` G ) ) )
4	3	biimpd 219	. . . 4 |- ( V = (Vtx ` G ) -> ( N e. V -> N e. (Vtx ` G ) ) )
5	4	eqcoms 2630	. . 3 |- ( (Vtx ` G ) = V -> ( N e. V -> N e. (Vtx ` G ) ) )
6	5	com12 32	. 2 |- ( N e. V -> ( (Vtx ` G ) = V -> N e. (Vtx ` G ) ) )
7	2, 6	mpan9 486	1 |- ( ( V e. W /\ N e. V ) -> N e. (Vtx ` G ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   <.cop 4183   ` cfv 5888  Vtxcvtx 25874

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-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-fv 5896  df-1st 7168  df-vtx 25876

This theorem is referenced by: (None)