Theorem upgredgpr 16073

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 16068	. . . 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 3840	. . . . . . . . . 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 3252	. . . . . . . . . 10 |- ( C = { a , b } -> ( { A , B } C_ C <-> { A , B } C_ { a , b } ) )
8		eqeq2 2241	. . . . . . . . . 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 2657	. . . . 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 2202   =/= wne 2403   E.wrex 2512   C_ wss 3201   {cpr 3674   ` cfv 5333  Vtxcvtx 15936  Edgcedg 15981  UPGraphcupgr 16015

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-1o 6625  df-2o 6626  df-en 6953  df-sub 8394  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-dec 9656  df-ndx 13148  df-slot 13149  df-base 13151  df-edgf 15929  df-vtx 15938  df-iedg 15939  df-edg 15982  df-upgren 16017

This theorem is referenced by:  upgriswlkdc  16284