Theorem umgrnloopvv 15760

Description: In a multigraph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 11-Dec-2020.)

Hypothesis

Ref	Expression
umgrnloopv.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
umgrnloopvv	|- ( ( G e. UMGraph /\ M e. W /\ N e. V ) -> ( ( E ` X ) = { M , N } -> M =/= N ) )

Proof of Theorem umgrnloopvv

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 |- ( ( ( G e. UMGraph /\ M e. W ) /\ ( E ` X ) = { M , N } ) -> ( E ` X ) = { M , N } )
2		simpll 527	. . . . . 6 |- ( ( ( G e. UMGraph /\ M e. W ) /\ ( E ` X ) = { M , N } ) -> G e. UMGraph )
3		umgruhgr 15759	. . . . . . . . 9 |- ( G e. UMGraph -> G e. UHGraph )
4		umgrnloopv.e	. . . . . . . . . 10 |- E = (iEdg ` G )
5	4	uhgrfun 15723	. . . . . . . . 9 |- ( G e. UHGraph -> Fun E )
6		funrel 5294	. . . . . . . . 9 |- ( Fun E -> Rel E )
7	3, 5, 6	3syl 17	. . . . . . . 8 |- ( G e. UMGraph -> Rel E )
8	7	ad2antrr 488	. . . . . . 7 |- ( ( ( G e. UMGraph /\ M e. W ) /\ ( E ` X ) = { M , N } ) -> Rel E )
9		simplr 528	. . . . . . . 8 |- ( ( ( G e. UMGraph /\ M e. W ) /\ ( E ` X ) = { M , N } ) -> M e. W )
10		prid1g 3739	. . . . . . . . . 10 |- ( M e. W -> M e. { M , N } )
11	10	adantl 277	. . . . . . . . 9 |- ( ( ( E ` X ) = { M , N } /\ M e. W ) -> M e. { M , N } )
12		eleq2 2270	. . . . . . . . . 10 |- ( ( E ` X ) = { M , N } -> ( M e. ( E ` X ) <-> M e. { M , N } ) )
13	12	adantr 276	. . . . . . . . 9 |- ( ( ( E ` X ) = { M , N } /\ M e. W ) -> ( M e. ( E ` X ) <-> M e. { M , N } ) )
14	11, 13	mpbird 167	. . . . . . . 8 |- ( ( ( E ` X ) = { M , N } /\ M e. W ) -> M e. ( E ` X ) )
15	1, 9, 14	syl2anc 411	. . . . . . 7 |- ( ( ( G e. UMGraph /\ M e. W ) /\ ( E ` X ) = { M , N } ) -> M e. ( E ` X ) )
16		relelfvdm 5618	. . . . . . 7 |- ( ( Rel E /\ M e. ( E ` X ) ) -> X e. dom E )
17	8, 15, 16	syl2anc 411	. . . . . 6 |- ( ( ( G e. UMGraph /\ M e. W ) /\ ( E ` X ) = { M , N } ) -> X e. dom E )
18		eqid 2206	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
19	18, 4	umgredg2en 15755	. . . . . 6 |- ( ( G e. UMGraph /\ X e. dom E ) -> ( E ` X ) ~~ 2o )
20	2, 17, 19	syl2anc 411	. . . . 5 |- ( ( ( G e. UMGraph /\ M e. W ) /\ ( E ` X ) = { M , N } ) -> ( E ` X ) ~~ 2o )
21	1, 20	eqbrtrrd 4072	. . . 4 |- ( ( ( G e. UMGraph /\ M e. W ) /\ ( E ` X ) = { M , N } ) -> { M , N } ~~ 2o )
22	21	3adantl3 1158	. . 3 |- ( ( ( G e. UMGraph /\ M e. W /\ N e. V ) /\ ( E ` X ) = { M , N } ) -> { M , N } ~~ 2o )
23		simpl2 1004	. . . 4 |- ( ( ( G e. UMGraph /\ M e. W /\ N e. V ) /\ ( E ` X ) = { M , N } ) -> M e. W )
24		simpl3 1005	. . . 4 |- ( ( ( G e. UMGraph /\ M e. W /\ N e. V ) /\ ( E ` X ) = { M , N } ) -> N e. V )
25		pr2ne 7312	. . . 4 |- ( ( M e. W /\ N e. V ) -> ( { M , N } ~~ 2o <-> M =/= N ) )
26	23, 24, 25	syl2anc 411	. . 3 |- ( ( ( G e. UMGraph /\ M e. W /\ N e. V ) /\ ( E ` X ) = { M , N } ) -> ( { M , N } ~~ 2o <-> M =/= N ) )
27	22, 26	mpbid 147	. 2 |- ( ( ( G e. UMGraph /\ M e. W /\ N e. V ) /\ ( E ` X ) = { M , N } ) -> M =/= N )
28	27	ex 115	1 |- ( ( G e. UMGraph /\ M e. W /\ N e. V ) -> ( ( E ` X ) = { M , N } -> M =/= N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2177   =/= wne 2377   {cpr 3636   class class class wbr 4048   dom cdm 4680   Rel wrel 4685   Fun wfun 5271   ` cfv 5277   2oc2o 6506   ~~ cen 6835  Vtxcvtx 15661  iEdgciedg 15662  UHGraphcuhgr 15713  UMGraphcumgr 15738

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-iinf 4641  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-addcom 8038  ax-mulcom 8039  ax-addass 8040  ax-mulass 8041  ax-distr 8042  ax-i2m1 8043  ax-1rid 8045  ax-0id 8046  ax-rnegex 8047  ax-cnre 8049

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-if 3574  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-br 4049  df-opab 4111  df-mpt 4112  df-tr 4148  df-id 4345  df-iord 4418  df-on 4420  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-1o 6512  df-2o 6513  df-er 6630  df-en 6838  df-sub 8258  df-inn 9050  df-2 9108  df-3 9109  df-4 9110  df-5 9111  df-6 9112  df-7 9113  df-8 9114  df-9 9115  df-n0 9309  df-dec 9518  df-ndx 12885  df-slot 12886  df-base 12888  df-edgf 15654  df-vtx 15663  df-iedg 15664  df-uhgrm 15715  df-upgren 15739  df-umgren 15740

This theorem is referenced by: (None)