Theorem upgrpredgv 16141

Description: An edge of a pseudograph always connects two vertices if the edge contains two sets. The two vertices/sets need not necessarily be different (loops are allowed). (Contributed by AV, 18-Nov-2021.)

Hypotheses

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

Assertion

Ref	Expression
upgrpredgv	|- ( ( G e. UPGraph /\ ( M e. U /\ N e. W ) /\ { M , N } e. E ) -> ( M e. V /\ N e. V ) )

Proof of Theorem upgrpredgv

Dummy variables m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgredg.v	. . . 4 |- V = (Vtx ` G )
2		upgredg.e	. . . 4 |- E = (Edg ` G )
3	1, 2	upgredg 16139	. . 3 |- ( ( G e. UPGraph /\ { M , N } e. E ) -> E. m e. V E. n e. V { M , N } = { m , n } )
4	3	3adant2 1043	. 2 |- ( ( G e. UPGraph /\ ( M e. U /\ N e. W ) /\ { M , N } e. E ) -> E. m e. V E. n e. V { M , N } = { m , n } )
5		preq12bg 3877	. . . . 5 |- ( ( ( M e. U /\ N e. W ) /\ ( m e. V /\ n e. V ) ) -> ( { M , N } = { m , n } <-> ( ( M = m /\ N = n ) \/ ( M = n /\ N = m ) ) ) )
6	5	3ad2antl2 1187	. . . 4 |- ( ( ( G e. UPGraph /\ ( M e. U /\ N e. W ) /\ { M , N } e. E ) /\ ( m e. V /\ n e. V ) ) -> ( { M , N } = { m , n } <-> ( ( M = m /\ N = n ) \/ ( M = n /\ N = m ) ) ) )
7		eleq1 2295	. . . . . . . . . 10 |- ( m = M -> ( m e. V <-> M e. V ) )
8	7	eqcoms 2235	. . . . . . . . 9 |- ( M = m -> ( m e. V <-> M e. V ) )
9	8	biimpd 144	. . . . . . . 8 |- ( M = m -> ( m e. V -> M e. V ) )
10		eleq1 2295	. . . . . . . . . 10 |- ( n = N -> ( n e. V <-> N e. V ) )
11	10	eqcoms 2235	. . . . . . . . 9 |- ( N = n -> ( n e. V <-> N e. V ) )
12	11	biimpd 144	. . . . . . . 8 |- ( N = n -> ( n e. V -> N e. V ) )
13	9, 12	im2anan9 602	. . . . . . 7 |- ( ( M = m /\ N = n ) -> ( ( m e. V /\ n e. V ) -> ( M e. V /\ N e. V ) ) )
14	13	com12 30	. . . . . 6 |- ( ( m e. V /\ n e. V ) -> ( ( M = m /\ N = n ) -> ( M e. V /\ N e. V ) ) )
15		eleq1 2295	. . . . . . . . . . 11 |- ( n = M -> ( n e. V <-> M e. V ) )
16	15	eqcoms 2235	. . . . . . . . . 10 |- ( M = n -> ( n e. V <-> M e. V ) )
17	16	biimpd 144	. . . . . . . . 9 |- ( M = n -> ( n e. V -> M e. V ) )
18		eleq1 2295	. . . . . . . . . . 11 |- ( m = N -> ( m e. V <-> N e. V ) )
19	18	eqcoms 2235	. . . . . . . . . 10 |- ( N = m -> ( m e. V <-> N e. V ) )
20	19	biimpd 144	. . . . . . . . 9 |- ( N = m -> ( m e. V -> N e. V ) )
21	17, 20	im2anan9 602	. . . . . . . 8 |- ( ( M = n /\ N = m ) -> ( ( n e. V /\ m e. V ) -> ( M e. V /\ N e. V ) ) )
22	21	com12 30	. . . . . . 7 |- ( ( n e. V /\ m e. V ) -> ( ( M = n /\ N = m ) -> ( M e. V /\ N e. V ) ) )
23	22	ancoms 268	. . . . . 6 |- ( ( m e. V /\ n e. V ) -> ( ( M = n /\ N = m ) -> ( M e. V /\ N e. V ) ) )
24	14, 23	jaod 725	. . . . 5 |- ( ( m e. V /\ n e. V ) -> ( ( ( M = m /\ N = n ) \/ ( M = n /\ N = m ) ) -> ( M e. V /\ N e. V ) ) )
25	24	adantl 277	. . . 4 |- ( ( ( G e. UPGraph /\ ( M e. U /\ N e. W ) /\ { M , N } e. E ) /\ ( m e. V /\ n e. V ) ) -> ( ( ( M = m /\ N = n ) \/ ( M = n /\ N = m ) ) -> ( M e. V /\ N e. V ) ) )
26	6, 25	sylbid 150	. . 3 |- ( ( ( G e. UPGraph /\ ( M e. U /\ N e. W ) /\ { M , N } e. E ) /\ ( m e. V /\ n e. V ) ) -> ( { M , N } = { m , n } -> ( M e. V /\ N e. V ) ) )
27	26	rexlimdvva 2668	. 2 |- ( ( G e. UPGraph /\ ( M e. U /\ N e. W ) /\ { M , N } e. E ) -> ( E. m e. V E. n e. V { M , N } = { m , n } -> ( M e. V /\ N e. V ) ) )
28	4, 27	mpd 13	1 |- ( ( G e. UPGraph /\ ( M e. U /\ N e. W ) /\ { M , N } e. E ) -> ( M e. V /\ N e. V ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2203   E.wrex 2521   {cpr 3690   ` cfv 5352  Vtxcvtx 16007  Edgcedg 16052  UPGraphcupgr 16086

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-1o 6647  df-2o 6648  df-en 6976  df-sub 8446  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-dec 9710  df-ndx 13215  df-slot 13216  df-base 13218  df-edgf 16000  df-vtx 16009  df-iedg 16010  df-edg 16053  df-upgren 16088

This theorem is referenced by: (None)