Theorem upgrpredgv 26034

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 26032	. . 3 |- ( ( G e. UPGraph /\ { M , N } e. E ) -> E. m e. V E. n e. V { M , N } = { m , n } )
4	3	3adant2 1080	. 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 4386	. . . . 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 1224	. . . 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 2689	. . . . . . . . . 10 |- ( m = M -> ( m e. V <-> M e. V ) )
8	7	eqcoms 2630	. . . . . . . . 9 |- ( M = m -> ( m e. V <-> M e. V ) )
9	8	biimpd 219	. . . . . . . 8 |- ( M = m -> ( m e. V -> M e. V ) )
10		eleq1 2689	. . . . . . . . . 10 |- ( n = N -> ( n e. V <-> N e. V ) )
11	10	eqcoms 2630	. . . . . . . . 9 |- ( N = n -> ( n e. V <-> N e. V ) )
12	11	biimpd 219	. . . . . . . 8 |- ( N = n -> ( n e. V -> N e. V ) )
13	9, 12	im2anan9 880	. . . . . . 7 |- ( ( M = m /\ N = n ) -> ( ( m e. V /\ n e. V ) -> ( M e. V /\ N e. V ) ) )
14	13	com12 32	. . . . . 6 |- ( ( m e. V /\ n e. V ) -> ( ( M = m /\ N = n ) -> ( M e. V /\ N e. V ) ) )
15		eleq1 2689	. . . . . . . . . . 11 |- ( n = M -> ( n e. V <-> M e. V ) )
16	15	eqcoms 2630	. . . . . . . . . 10 |- ( M = n -> ( n e. V <-> M e. V ) )
17	16	biimpd 219	. . . . . . . . 9 |- ( M = n -> ( n e. V -> M e. V ) )
18		eleq1 2689	. . . . . . . . . . 11 |- ( m = N -> ( m e. V <-> N e. V ) )
19	18	eqcoms 2630	. . . . . . . . . 10 |- ( N = m -> ( m e. V <-> N e. V ) )
20	19	biimpd 219	. . . . . . . . 9 |- ( N = m -> ( m e. V -> N e. V ) )
21	17, 20	im2anan9 880	. . . . . . . 8 |- ( ( M = n /\ N = m ) -> ( ( n e. V /\ m e. V ) -> ( M e. V /\ N e. V ) ) )
22	21	com12 32	. . . . . . 7 |- ( ( n e. V /\ m e. V ) -> ( ( M = n /\ N = m ) -> ( M e. V /\ N e. V ) ) )
23	22	ancoms 469	. . . . . 6 |- ( ( m e. V /\ n e. V ) -> ( ( M = n /\ N = m ) -> ( M e. V /\ N e. V ) ) )
24	14, 23	jaod 395	. . . . 5 |- ( ( m e. V /\ n e. V ) -> ( ( ( M = m /\ N = n ) \/ ( M = n /\ N = m ) ) -> ( M e. V /\ N e. V ) ) )
25	24	adantl 482	. . . 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 230	. . 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 3038	. 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 15	1 |- ( ( G e. UPGraph /\ ( M e. U /\ N e. W ) /\ { M , N } e. E ) -> ( M e. V /\ N e. V ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   {cpr 4179   ` cfv 5888  Vtxcvtx 25874  Edgcedg 25939   UPGraph cupgr 25975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-upgr 25977

This theorem is referenced by: (None)