Theorem umgrvad2edg 16017

Description: If a vertex is adjacent to two different vertices in a multigraph, there are more than one edges starting at this vertex, analogous to usgr2edg 16014. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 8-Jun-2021.)

Hypothesis

Ref	Expression
umgrvad2edg.e	|- E = (Edg ` G )

Assertion

Ref	Expression
umgrvad2edg	|- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> E. x e. E E. y e. E ( x =/= y /\ N e. x /\ N e. y ) )

Distinct variable groups:   x, A, y   x, B, y   x, E, y   x, G, y   x, N, y

Proof of Theorem umgrvad2edg

Step	Hyp	Ref	Expression
1		simpl 109	. 2 |- ( ( { N , A } e. E /\ { B , N } e. E ) -> { N , A } e. E )
2		simpr 110	. 2 |- ( ( { N , A } e. E /\ { B , N } e. E ) -> { B , N } e. E )
3		eqid 2229	. . . . . . . 8 |- (Vtx ` G ) = (Vtx ` G )
4		umgrvad2edg.e	. . . . . . . 8 |- E = (Edg ` G )
5	3, 4	umgrpredgv 15953	. . . . . . 7 |- ( ( G e. UMGraph /\ { N , A } e. E ) -> ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) )
6	5	ex 115	. . . . . 6 |- ( G e. UMGraph -> ( { N , A } e. E -> ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) ) )
7	3, 4	umgrpredgv 15953	. . . . . . 7 |- ( ( G e. UMGraph /\ { B , N } e. E ) -> ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) )
8	7	ex 115	. . . . . 6 |- ( G e. UMGraph -> ( { B , N } e. E -> ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) )
9	6, 8	anim12d 335	. . . . 5 |- ( G e. UMGraph -> ( ( { N , A } e. E /\ { B , N } e. E ) -> ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) ) )
10	9	adantr 276	. . . 4 |- ( ( G e. UMGraph /\ A =/= B ) -> ( ( { N , A } e. E /\ { B , N } e. E ) -> ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) ) )
11	10	imp 124	. . 3 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) )
12		simplr 528	. . . . 5 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> A =/= B )
13	4	umgredgne 15956	. . . . . . 7 |- ( ( G e. UMGraph /\ { N , A } e. E ) -> N =/= A )
14	13	necomd 2486	. . . . . 6 |- ( ( G e. UMGraph /\ { N , A } e. E ) -> A =/= N )
15	14	ad2ant2r 509	. . . . 5 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> A =/= N )
16	12, 15	jca 306	. . . 4 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( A =/= B /\ A =/= N ) )
17	16	olcd 739	. . 3 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( ( N =/= B /\ N =/= N ) \/ ( A =/= B /\ A =/= N ) ) )
18		prneimg 3852	. . . . 5 |- ( ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) -> ( ( ( N =/= B /\ N =/= N ) \/ ( A =/= B /\ A =/= N ) ) -> { N , A } =/= { B , N } ) )
19	18	imp 124	. . . 4 |- ( ( ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) /\ ( ( N =/= B /\ N =/= N ) \/ ( A =/= B /\ A =/= N ) ) ) -> { N , A } =/= { B , N } )
20		prid1g 3770	. . . . 5 |- ( N e. (Vtx ` G ) -> N e. { N , A } )
21	20	ad3antrrr 492	. . . 4 |- ( ( ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) /\ ( ( N =/= B /\ N =/= N ) \/ ( A =/= B /\ A =/= N ) ) ) -> N e. { N , A } )
22		prid2g 3771	. . . . 5 |- ( N e. (Vtx ` G ) -> N e. { B , N } )
23	22	ad3antrrr 492	. . . 4 |- ( ( ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) /\ ( ( N =/= B /\ N =/= N ) \/ ( A =/= B /\ A =/= N ) ) ) -> N e. { B , N } )
24	19, 21, 23	3jca 1201	. . 3 |- ( ( ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) /\ ( ( N =/= B /\ N =/= N ) \/ ( A =/= B /\ A =/= N ) ) ) -> ( { N , A } =/= { B , N } /\ N e. { N , A } /\ N e. { B , N } ) )
25	11, 17, 24	syl2anc 411	. 2 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( { N , A } =/= { B , N } /\ N e. { N , A } /\ N e. { B , N } ) )
26		neeq1 2413	. . . 4 |- ( x = { N , A } -> ( x =/= y <-> { N , A } =/= y ) )
27		eleq2 2293	. . . 4 |- ( x = { N , A } -> ( N e. x <-> N e. { N , A } ) )
28	26, 27	3anbi12d 1347	. . 3 |- ( x = { N , A } -> ( ( x =/= y /\ N e. x /\ N e. y ) <-> ( { N , A } =/= y /\ N e. { N , A } /\ N e. y ) ) )
29		neeq2 2414	. . . 4 |- ( y = { B , N } -> ( { N , A } =/= y <-> { N , A } =/= { B , N } ) )
30		eleq2 2293	. . . 4 |- ( y = { B , N } -> ( N e. y <-> N e. { B , N } ) )
31	29, 30	3anbi13d 1348	. . 3 |- ( y = { B , N } -> ( ( { N , A } =/= y /\ N e. { N , A } /\ N e. y ) <-> ( { N , A } =/= { B , N } /\ N e. { N , A } /\ N e. { B , N } ) ) )
32	28, 31	rspc2ev 2922	. 2 |- ( ( { N , A } e. E /\ { B , N } e. E /\ ( { N , A } =/= { B , N } /\ N e. { N , A } /\ N e. { B , N } ) ) -> E. x e. E E. y e. E ( x =/= y /\ N e. x /\ N e. y ) )
33	1, 2, 25, 32	syl2an23an 1333	1 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> E. x e. E E. y e. E ( x =/= y /\ N e. x /\ N e. y ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   E.wrex 2509   {cpr 3667   ` cfv 5318  Vtxcvtx 15821  Edgcedg 15866  UMGraphcumgr 15900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-cnre 8118

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-1o 6568  df-2o 6569  df-er 6688  df-en 6896  df-sub 8327  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-7 9182  df-8 9183  df-9 9184  df-n0 9378  df-dec 9587  df-ndx 13043  df-slot 13044  df-base 13046  df-edgf 15814  df-vtx 15823  df-iedg 15824  df-edg 15867  df-umgren 15902

This theorem is referenced by:  umgr2edgneu  16018