Theorem upgrpredgv 16267

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 16265	. . 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 3882	. . . . 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 2297	. . . . . . . . . 10 |- ( m = M -> ( m e. V <-> M e. V ) )
8	7	eqcoms 2237	. . . . . . . . 9 |- ( M = m -> ( m e. V <-> M e. V ) )
9	8	biimpd 144	. . . . . . . 8 |- ( M = m -> ( m e. V -> M e. V ) )
10		eleq1 2297	. . . . . . . . . 10 |- ( n = N -> ( n e. V <-> N e. V ) )
11	10	eqcoms 2237	. . . . . . . . 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 2297	. . . . . . . . . . 11 |- ( n = M -> ( n e. V <-> M e. V ) )
16	15	eqcoms 2237	. . . . . . . . . 10 |- ( M = n -> ( n e. V <-> M e. V ) )
17	16	biimpd 144	. . . . . . . . 9 |- ( M = n -> ( n e. V -> M e. V ) )
18		eleq1 2297	. . . . . . . . . . 11 |- ( m = N -> ( m e. V <-> N e. V ) )
19	18	eqcoms 2237	. . . . . . . . . 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 2670	. 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 2205   E.wrex 2523   {cpr 3695   ` cfv 5357  Vtxcvtx 16133  Edgcedg 16178  UPGraphcupgr 16212

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-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-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-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-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-upgren 16214

This theorem is referenced by: (None)