Theorem frcond1 27130

Description: The friendship condition: any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.)

Hypotheses

Ref	Expression
frcond1.v	|- V = (Vtx ` G )
frcond1.e	|- E = (Edg ` G )

Assertion

Ref	Expression
frcond1	|- ( G e. FriendGraph -> ( ( A e. V /\ C e. V /\ A =/= C ) -> E! b e. V { { A , b } , { b , C } } C_ E ) )

Distinct variable groups:   A, b   C, b   G, b   V, b

Allowed substitution hint:   E( b)

Proof of Theorem frcond1

Dummy variables k l are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frcond1.v	. . 3 |- V = (Vtx ` G )
2		frcond1.e	. . 3 |- E = (Edg ` G )
3	1, 2	frgrusgrfrcond 27123	. 2 |- ( G e. FriendGraph <-> ( G e. USGraph /\ A. k e. V A. l e. ( V \ { k } ) E! b e. V { { b , k } , { b , l } } C_ E ) )
4		preq2 4269	. . . . . . 7 |- ( k = A -> { b , k } = { b , A } )
5	4	preq1d 4274	. . . . . 6 |- ( k = A -> { { b , k } , { b , l } } = { { b , A } , { b , l } } )
6	5	sseq1d 3632	. . . . 5 |- ( k = A -> ( { { b , k } , { b , l } } C_ E <-> { { b , A } , { b , l } } C_ E ) )
7	6	reubidv 3126	. . . 4 |- ( k = A -> ( E! b e. V { { b , k } , { b , l } } C_ E <-> E! b e. V { { b , A } , { b , l } } C_ E ) )
8		preq2 4269	. . . . . . 7 |- ( l = C -> { b , l } = { b , C } )
9	8	preq2d 4275	. . . . . 6 |- ( l = C -> { { b , A } , { b , l } } = { { b , A } , { b , C } } )
10	9	sseq1d 3632	. . . . 5 |- ( l = C -> ( { { b , A } , { b , l } } C_ E <-> { { b , A } , { b , C } } C_ E ) )
11	10	reubidv 3126	. . . 4 |- ( l = C -> ( E! b e. V { { b , A } , { b , l } } C_ E <-> E! b e. V { { b , A } , { b , C } } C_ E ) )
12		simp1 1061	. . . 4 |- ( ( A e. V /\ C e. V /\ A =/= C ) -> A e. V )
13		sneq 4187	. . . . . 6 |- ( k = A -> { k } = { A } )
14	13	difeq2d 3728	. . . . 5 |- ( k = A -> ( V \ { k } ) = ( V \ { A } ) )
15	14	adantl 482	. . . 4 |- ( ( ( A e. V /\ C e. V /\ A =/= C ) /\ k = A ) -> ( V \ { k } ) = ( V \ { A } ) )
16		necom 2847	. . . . . . . 8 |- ( A =/= C <-> C =/= A )
17	16	biimpi 206	. . . . . . 7 |- ( A =/= C -> C =/= A )
18	17	anim2i 593	. . . . . 6 |- ( ( C e. V /\ A =/= C ) -> ( C e. V /\ C =/= A ) )
19	18	3adant1 1079	. . . . 5 |- ( ( A e. V /\ C e. V /\ A =/= C ) -> ( C e. V /\ C =/= A ) )
20		eldifsn 4317	. . . . 5 |- ( C e. ( V \ { A } ) <-> ( C e. V /\ C =/= A ) )
21	19, 20	sylibr 224	. . . 4 |- ( ( A e. V /\ C e. V /\ A =/= C ) -> C e. ( V \ { A } ) )
22	7, 11, 12, 15, 21	rspc2vd 27129	. . 3 |- ( ( A e. V /\ C e. V /\ A =/= C ) -> ( A. k e. V A. l e. ( V \ { k } ) E! b e. V { { b , k } , { b , l } } C_ E -> E! b e. V { { b , A } , { b , C } } C_ E ) )
23		prcom 4267	. . . . . . 7 |- { b , A } = { A , b }
24	23	preq1i 4271	. . . . . 6 |- { { b , A } , { b , C } } = { { A , b } , { b , C } }
25	24	sseq1i 3629	. . . . 5 |- ( { { b , A } , { b , C } } C_ E <-> { { A , b } , { b , C } } C_ E )
26	25	reubii 3128	. . . 4 |- ( E! b e. V { { b , A } , { b , C } } C_ E <-> E! b e. V { { A , b } , { b , C } } C_ E )
27	26	biimpi 206	. . 3 |- ( E! b e. V { { b , A } , { b , C } } C_ E -> E! b e. V { { A , b } , { b , C } } C_ E )
28	22, 27	syl6com 37	. 2 |- ( A. k e. V A. l e. ( V \ { k } ) E! b e. V { { b , k } , { b , l } } C_ E -> ( ( A e. V /\ C e. V /\ A =/= C ) -> E! b e. V { { A , b } , { b , C } } C_ E ) )
29	3, 28	simplbiim 659	1 |- ( G e. FriendGraph -> ( ( A e. V /\ C e. V /\ A =/= C ) -> E! b e. V { { A , b } , { b , C } } C_ E ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E!wreu 2914   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   ` cfv 5888  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   FriendGraph cfrgr 27120

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-frgr 27121

This theorem is referenced by:  frcond2  27131  frcond3  27133  4cyclusnfrgr  27156  frgrncvvdeqlem2  27164