Theorem upgredgpr 16144

Description: If a proper pair (of vertices) is a subset of an edge in a pseudograph, the pair is the edge. (Contributed by AV, 30-Dec-2020.)

Hypotheses

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

Assertion

Ref	Expression
upgredgpr	|- ( ( ( G e. UPGraph /\ C e. E /\ { A , B } C_ C ) /\ ( A e. U /\ B e. W /\ A =/= B ) ) -> { A , B } = C )

Proof of Theorem upgredgpr

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgredg.v	. . . . 5 |- V = (Vtx ` G )
2		upgredg.e	. . . . 5 |- E = (Edg ` G )
3	1, 2	upgredg 16139	. . . 4 |- ( ( G e. UPGraph /\ C e. E ) -> E. a e. V E. b e. V C = { a , b } )
4	3	3adant3 1044	. . 3 |- ( ( G e. UPGraph /\ C e. E /\ { A , B } C_ C ) -> E. a e. V E. b e. V C = { a , b } )
5		ssprsseq 3856	. . . . . . . . . 10 |- ( ( A e. U /\ B e. W /\ A =/= B ) -> ( { A , B } C_ { a , b } <-> { A , B } = { a , b } ) )
6	5	biimpd 144	. . . . . . . . 9 |- ( ( A e. U /\ B e. W /\ A =/= B ) -> ( { A , B } C_ { a , b } -> { A , B } = { a , b } ) )
7		sseq2 3262	. . . . . . . . . 10 |- ( C = { a , b } -> ( { A , B } C_ C <-> { A , B } C_ { a , b } ) )
8		eqeq2 2242	. . . . . . . . . 10 |- ( C = { a , b } -> ( { A , B } = C <-> { A , B } = { a , b } ) )
9	7, 8	imbi12d 234	. . . . . . . . 9 |- ( C = { a , b } -> ( ( { A , B } C_ C -> { A , B } = C ) <-> ( { A , B } C_ { a , b } -> { A , B } = { a , b } ) ) )
10	6, 9	imbitrrid 156	. . . . . . . 8 |- ( C = { a , b } -> ( ( A e. U /\ B e. W /\ A =/= B ) -> ( { A , B } C_ C -> { A , B } = C ) ) )
11	10	com23 78	. . . . . . 7 |- ( C = { a , b } -> ( { A , B } C_ C -> ( ( A e. U /\ B e. W /\ A =/= B ) -> { A , B } = C ) ) )
12	11	a1i 9	. . . . . 6 |- ( ( a e. V /\ b e. V ) -> ( C = { a , b } -> ( { A , B } C_ C -> ( ( A e. U /\ B e. W /\ A =/= B ) -> { A , B } = C ) ) ) )
13	12	rexlimivv 2666	. . . . 5 |- ( E. a e. V E. b e. V C = { a , b } -> ( { A , B } C_ C -> ( ( A e. U /\ B e. W /\ A =/= B ) -> { A , B } = C ) ) )
14	13	com12 30	. . . 4 |- ( { A , B } C_ C -> ( E. a e. V E. b e. V C = { a , b } -> ( ( A e. U /\ B e. W /\ A =/= B ) -> { A , B } = C ) ) )
15	14	3ad2ant3 1047	. . 3 |- ( ( G e. UPGraph /\ C e. E /\ { A , B } C_ C ) -> ( E. a e. V E. b e. V C = { a , b } -> ( ( A e. U /\ B e. W /\ A =/= B ) -> { A , B } = C ) ) )
16	4, 15	mpd 13	. 2 |- ( ( G e. UPGraph /\ C e. E /\ { A , B } C_ C ) -> ( ( A e. U /\ B e. W /\ A =/= B ) -> { A , B } = C ) )
17	16	imp 124	1 |- ( ( ( G e. UPGraph /\ C e. E /\ { A , B } C_ C ) /\ ( A e. U /\ B e. W /\ A =/= B ) ) -> { A , B } = C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   E.wrex 2521   C_ wss 3211   {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:  upgriswlkdc  16355