Theorem umgrvad2edg 16435

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 16432. (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 2238	. . . . . . . 8 |- (Vtx ` G ) = (Vtx ` G )
4		umgrvad2edg.e	. . . . . . . 8 |- E = (Edg ` G )
5	3, 4	umgrpredgv 16371	. . . . . . 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 16371	. . . . . . 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 533	. . . . 5 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> A =/= B )
13	4	umgredgne 16374	. . . . . . 7 |- ( ( G e. UMGraph /\ { N , A } e. E ) -> N =/= A )
14	13	necomd 2506	. . . . . 6 |- ( ( G e. UMGraph /\ { N , A } e. E ) -> A =/= N )
15	14	ad2ant2r 513	. . . . 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 746	. . 3 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( ( N =/= B /\ N =/= N ) \/ ( A =/= B /\ A =/= N ) ) )
18		prneimg 3897	. . . . 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 3814	. . . . 5 |- ( N e. (Vtx ` G ) -> N e. { N , A } )
21	20	ad3antrrr 496	. . . 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 3815	. . . . 5 |- ( N e. (Vtx ` G ) -> N e. { B , N } )
23	22	ad3antrrr 496	. . . 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 1208	. . 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 415	. 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 2433	. . . 4 |- ( x = { N , A } -> ( x =/= y <-> { N , A } =/= y ) )
27		eleq2 2302	. . . 4 |- ( x = { N , A } -> ( N e. x <-> N e. { N , A } ) )
28	26, 27	3anbi12d 1354	. . 3 |- ( x = { N , A } -> ( ( x =/= y /\ N e. x /\ N e. y ) <-> ( { N , A } =/= y /\ N e. { N , A } /\ N e. y ) ) )
29		neeq2 2434	. . . 4 |- ( y = { B , N } -> ( { N , A } =/= y <-> { N , A } =/= { B , N } ) )
30		eleq2 2302	. . . 4 |- ( y = { B , N } -> ( N e. y <-> N e. { B , N } ) )
31	29, 30	3anbi13d 1355	. . 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 2945	. 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 1340	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 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   E.wrex 2529   {cpr 3709   ` cfv 5375  Vtxcvtx 16236  Edgcedg 16281  UMGraphcumgr 16316

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-1o 6681  df-2o 6682  df-er 6801  df-en 7017  df-sub 8493  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-dec 9761  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-edg 16282  df-umgren 16318

This theorem is referenced by:  umgr2edgneu  16436