Theorem upgrpredgv 15985

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 15983	. . 3 |- ( ( G e. UPGraph /\ { M , N } e. E ) -> E. m e. V E. n e. V { M , N } = { m , n } )
4	3	3adant2 1040	. 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 3854	. . . . 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 1184	. . . 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 2292	. . . . . . . . . 10 |- ( m = M -> ( m e. V <-> M e. V ) )
8	7	eqcoms 2232	. . . . . . . . 9 |- ( M = m -> ( m e. V <-> M e. V ) )
9	8	biimpd 144	. . . . . . . 8 |- ( M = m -> ( m e. V -> M e. V ) )
10		eleq1 2292	. . . . . . . . . 10 |- ( n = N -> ( n e. V <-> N e. V ) )
11	10	eqcoms 2232	. . . . . . . . 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 600	. . . . . . 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 2292	. . . . . . . . . . 11 |- ( n = M -> ( n e. V <-> M e. V ) )
16	15	eqcoms 2232	. . . . . . . . . 10 |- ( M = n -> ( n e. V <-> M e. V ) )
17	16	biimpd 144	. . . . . . . . 9 |- ( M = n -> ( n e. V -> M e. V ) )
18		eleq1 2292	. . . . . . . . . . 11 |- ( m = N -> ( m e. V <-> N e. V ) )
19	18	eqcoms 2232	. . . . . . . . . 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 600	. . . . . . . 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 722	. . . . 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 2656	. 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 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   {cpr 3668   ` cfv 5324  Vtxcvtx 15853  Edgcedg 15898  UPGraphcupgr 15932

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-1o 6577  df-2o 6578  df-en 6905  df-sub 8342  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-9 9199  df-n0 9393  df-dec 9602  df-ndx 13075  df-slot 13076  df-base 13078  df-edgf 15846  df-vtx 15855  df-iedg 15856  df-edg 15899  df-upgren 15934

This theorem is referenced by: (None)