Theorem umgredg 16266

Description: For each edge in a multigraph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 25-Nov-2020.)

Hypotheses

Ref	Expression
upgredg.v	|- V = (Vtx ` G )
upgredg.e	|- E = (Edg ` G )

Assertion

Ref	Expression
umgredg	|- ( ( G e. UMGraph /\ C e. E ) -> E. a e. V E. b e. V ( a =/= b /\ C = { a , b } ) )

Distinct variable groups:   C, a, b   G, a, b   V, a, b

Allowed substitution hints:   E( a, b)

Proof of Theorem umgredg

Step	Hyp	Ref	Expression
1		upgredg.e	. . . . 5 |- E = (Edg ` G )
2	1	eleq2i 2301	. . . 4 |- ( C e. E <-> C e. (Edg ` G ) )
3		edgumgren 16263	. . . 4 |- ( ( G e. UMGraph /\ C e. (Edg ` G ) ) -> ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) )
4	2, 3	sylan2b 287	. . 3 |- ( ( G e. UMGraph /\ C e. E ) -> ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) )
5		en2prde 7503	. . . . 5 |- ( C ~~ 2o -> E. a E. b ( a =/= b /\ C = { a , b } ) )
6	5	adantl 277	. . . 4 |- ( ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) -> E. a E. b ( a =/= b /\ C = { a , b } ) )
7		eleq1 2297	. . . . . . . . . 10 |- ( C = { a , b } -> ( C e. ~P (Vtx ` G ) <-> { a , b } e. ~P (Vtx ` G ) ) )
8		zfpair2 4328	. . . . . . . . . . . 12 |- { a , b } e. _V
9	8	elpw 3680	. . . . . . . . . . 11 |- ( { a , b } e. ~P (Vtx ` G ) <-> { a , b } C_ (Vtx ` G ) )
10		vex 2818	. . . . . . . . . . . . 13 |- a e. _V
11		vex 2818	. . . . . . . . . . . . 13 |- b e. _V
12	10, 11	prss 3855	. . . . . . . . . . . 12 |- ( ( a e. V /\ b e. V ) <-> { a , b } C_ V )
13		upgredg.v	. . . . . . . . . . . . 13 |- V = (Vtx ` G )
14	13	sseq2i 3269	. . . . . . . . . . . 12 |- ( { a , b } C_ V <-> { a , b } C_ (Vtx ` G ) )
15	12, 14	sylbbr 136	. . . . . . . . . . 11 |- ( { a , b } C_ (Vtx ` G ) -> ( a e. V /\ b e. V ) )
16	9, 15	sylbi 121	. . . . . . . . . 10 |- ( { a , b } e. ~P (Vtx ` G ) -> ( a e. V /\ b e. V ) )
17	7, 16	biimtrdi 163	. . . . . . . . 9 |- ( C = { a , b } -> ( C e. ~P (Vtx ` G ) -> ( a e. V /\ b e. V ) ) )
18	17	adantrd 279	. . . . . . . 8 |- ( C = { a , b } -> ( ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) -> ( a e. V /\ b e. V ) ) )
19	18	adantl 277	. . . . . . 7 |- ( ( a =/= b /\ C = { a , b } ) -> ( ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) -> ( a e. V /\ b e. V ) ) )
20	19	imdistanri 446	. . . . . 6 |- ( ( ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) /\ ( a =/= b /\ C = { a , b } ) ) -> ( ( a e. V /\ b e. V ) /\ ( a =/= b /\ C = { a , b } ) ) )
21	20	ex 115	. . . . 5 |- ( ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) -> ( ( a =/= b /\ C = { a , b } ) -> ( ( a e. V /\ b e. V ) /\ ( a =/= b /\ C = { a , b } ) ) ) )
22	21	2eximdv 1931	. . . 4 |- ( ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) -> ( E. a E. b ( a =/= b /\ C = { a , b } ) -> E. a E. b ( ( a e. V /\ b e. V ) /\ ( a =/= b /\ C = { a , b } ) ) ) )
23	6, 22	mpd 13	. . 3 |- ( ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) -> E. a E. b ( ( a e. V /\ b e. V ) /\ ( a =/= b /\ C = { a , b } ) ) )
24	4, 23	syl 14	. 2 |- ( ( G e. UMGraph /\ C e. E ) -> E. a E. b ( ( a e. V /\ b e. V ) /\ ( a =/= b /\ C = { a , b } ) ) )
25		r2ex 2564	. 2 |- ( E. a e. V E. b e. V ( a =/= b /\ C = { a , b } ) <-> E. a E. b ( ( a e. V /\ b e. V ) /\ ( a =/= b /\ C = { a , b } ) ) )
26	24, 25	sylibr 134	1 |- ( ( G e. UMGraph /\ C e. E ) -> E. a e. V E. b e. V ( a =/= b /\ C = { a , b } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2205   =/= wne 2414   E.wrex 2523   C_ wss 3214   ~Pcpw 3674   {cpr 3695   class class class wbr 4114   ` cfv 5357   2oc2o 6654   ~~ cen 6986  Vtxcvtx 16133  Edgcedg 16178  UMGraphcumgr 16213

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 8462  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-dec 9728  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-edg 16179  df-umgren 16215

This theorem is referenced by:  usgredg  16321