Theorem umgredg 16300

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 2305	. . . 4 |- ( C e. E <-> C e. (Edg ` G ) )
3		edgumgren 16297	. . . 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 7529	. . . . 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 2301	. . . . . . . . . 10 |- ( C = { a , b } -> ( C e. ~P (Vtx ` G ) <-> { a , b } e. ~P (Vtx ` G ) ) )
8		zfpair2 4342	. . . . . . . . . . . 12 |- { a , b } e. _V
9	8	elpw 3691	. . . . . . . . . . 11 |- ( { a , b } e. ~P (Vtx ` G ) <-> { a , b } C_ (Vtx ` G ) )
10		vex 2824	. . . . . . . . . . . . 13 |- a e. _V
11		vex 2824	. . . . . . . . . . . . 13 |- b e. _V
12	10, 11	prss 3866	. . . . . . . . . . . 12 |- ( ( a e. V /\ b e. V ) <-> { a , b } C_ V )
13		upgredg.v	. . . . . . . . . . . . 13 |- V = (Vtx ` G )
14	13	sseq2i 3275	. . . . . . . . . . . 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 450	. . . . . 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 1935	. . . 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 2570	. 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 1402   E.wex 1545   e. wcel 2209   =/= wne 2420   E.wrex 2529   C_ wss 3220   ~Pcpw 3685   {cpr 3706   class class class wbr 4125   ` cfv 5372   2oc2o 6671   ~~ cen 7010  Vtxcvtx 16167  Edgcedg 16212  UMGraphcumgr 16247

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 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-1o 6677  df-2o 6678  df-er 6797  df-en 7013  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-edg 16213  df-umgren 16249

This theorem is referenced by:  usgredg  16355