Theorem umgrvad2edg 16335

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 16332. (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 2234	. . . . . . . 8 |- (Vtx ` G ) = (Vtx ` G )
4		umgrvad2edg.e	. . . . . . . 8 |- E = (Edg ` G )
5	3, 4	umgrpredgv 16271	. . . . . . 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 16271	. . . . . . 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 529	. . . . 5 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> A =/= B )
13	4	umgredgne 16274	. . . . . . 7 |- ( ( G e. UMGraph /\ { N , A } e. E ) -> N =/= A )
14	13	necomd 2500	. . . . . 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 742	. . 3 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( ( N =/= B /\ N =/= N ) \/ ( A =/= B /\ A =/= N ) ) )
18		prneimg 3883	. . . . 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 3800	. . . . 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 3801	. . . . 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 1204	. . 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 2427	. . . 4 |- ( x = { N , A } -> ( x =/= y <-> { N , A } =/= y ) )
27		eleq2 2298	. . . 4 |- ( x = { N , A } -> ( N e. x <-> N e. { N , A } ) )
28	26, 27	3anbi12d 1350	. . 3 |- ( x = { N , A } -> ( ( x =/= y /\ N e. x /\ N e. y ) <-> ( { N , A } =/= y /\ N e. { N , A } /\ N e. y ) ) )
29		neeq2 2428	. . . 4 |- ( y = { B , N } -> ( { N , A } =/= y <-> { N , A } =/= { B , N } ) )
30		eleq2 2298	. . . 4 |- ( y = { B , N } -> ( N e. y <-> N e. { B , N } ) )
31	29, 30	3anbi13d 1351	. . 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 2939	. 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 1336	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 716   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   E.wrex 2523   {cpr 3695   ` cfv 5357  Vtxcvtx 16136  Edgcedg 16181  UMGraphcumgr 16216

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-1o 6660  df-2o 6661  df-er 6780  df-en 6989  df-sub 8463  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-5 9319  df-6 9320  df-7 9321  df-8 9322  df-9 9323  df-n0 9517  df-dec 9731  df-ndx 13302  df-slot 13303  df-base 13305  df-edgf 16129  df-vtx 16138  df-iedg 16139  df-edg 16182  df-umgren 16218

This theorem is referenced by:  umgr2edgneu  16336