Theorem upgredgpr 16019

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 16014	. . . 4 |- ( ( G e. UPGraph /\ C e. E ) -> E. a e. V E. b e. V C = { a , b } )
4	3	3adant3 1043	. . 3 |- ( ( G e. UPGraph /\ C e. E /\ { A , B } C_ C ) -> E. a e. V E. b e. V C = { a , b } )
5		ssprsseq 3835	. . . . . . . . . 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 3251	. . . . . . . . . 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 2656	. . . . 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 1046	. . 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 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   E.wrex 2511   C_ wss 3200   {cpr 3670   ` cfv 5326  Vtxcvtx 15882  Edgcedg 15927  UPGraphcupgr 15961

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-1o 6582  df-2o 6583  df-en 6910  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13103  df-slot 13104  df-base 13106  df-edgf 15875  df-vtx 15884  df-iedg 15885  df-edg 15928  df-upgren 15963

This theorem is referenced by:  upgriswlkdc  16230