Theorem 4cycl2v2nb 27153

Description: In a (maybe degenerated) 4-cycle, two vertice have two (maybe not different) common neighbors. (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.)

Assertion

Ref	Expression
4cycl2v2nb	|- ( ( ( { A , B } e. E /\ { B , C } e. E ) /\ ( { C , D } e. E /\ { D , A } e. E ) ) -> ( { { A , B } , { B , C } } C_ E /\ { { A , D } , { D , C } } C_ E ) )

Proof of Theorem 4cycl2v2nb

Step	Hyp	Ref	Expression
1		prssi 4353	. 2 |- ( ( { A , B } e. E /\ { B , C } e. E ) -> { { A , B } , { B , C } } C_ E )
2		prcom 4267	. . . . 5 |- { D , A } = { A , D }
3	2	eleq1i 2692	. . . 4 |- ( { D , A } e. E <-> { A , D } e. E )
4	3	biimpi 206	. . 3 |- ( { D , A } e. E -> { A , D } e. E )
5		prcom 4267	. . . . 5 |- { C , D } = { D , C }
6	5	eleq1i 2692	. . . 4 |- ( { C , D } e. E <-> { D , C } e. E )
7	6	biimpi 206	. . 3 |- ( { C , D } e. E -> { D , C } e. E )
8		prssi 4353	. . 3 |- ( ( { A , D } e. E /\ { D , C } e. E ) -> { { A , D } , { D , C } } C_ E )
9	4, 7, 8	syl2anr 495	. 2 |- ( ( { C , D } e. E /\ { D , A } e. E ) -> { { A , D } , { D , C } } C_ E )
10	1, 9	anim12i 590	1 |- ( ( ( { A , B } e. E /\ { B , C } e. E ) /\ ( { C , D } e. E /\ { D , A } e. E ) ) -> ( { { A , B } , { B , C } } C_ E /\ { { A , D } , { D , C } } C_ E ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   C_ wss 3574   {cpr 4179

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-pr 4180

This theorem is referenced by:  4cycl2vnunb  27154