Theorem frgrwopreglem3 27178

Description: Lemma 3 for frgrwopreg 27187. The vertices in the sets A and B have different degrees. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 2-Jan-2022.)

Hypotheses

Ref	Expression
frgrwopreg.v	|- V = (Vtx ` G )
frgrwopreg.d	|- D = (VtxDeg ` G )
frgrwopreg.a	|- A = { x e. V | ( D ` x ) = K }
frgrwopreg.b	|- B = ( V \ A )

Assertion

Ref	Expression
frgrwopreglem3	|- ( ( X e. A /\ Y e. B ) -> ( D ` X ) =/= ( D ` Y ) )

Distinct variable groups:   x, V   x, A   x, G   x, K   x, D   x, X   x, Y

Allowed substitution hint:   B( x)

Proof of Theorem frgrwopreglem3

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( x = Y -> ( D ` x ) = ( D ` Y ) )
2	1	eqeq1d 2624	. . . . 5 |- ( x = Y -> ( ( D ` x ) = K <-> ( D ` Y ) = K ) )
3	2	notbid 308	. . . 4 |- ( x = Y -> ( -. ( D ` x ) = K <-> -. ( D ` Y ) = K ) )
4		frgrwopreg.b	. . . . 5 |- B = ( V \ A )
5		frgrwopreg.a	. . . . . 6 |- A = { x e. V | ( D ` x ) = K }
6	5	difeq2i 3725	. . . . 5 |- ( V \ A ) = ( V \ { x e. V | ( D ` x ) = K } )
7		notrab 3904	. . . . 5 |- ( V \ { x e. V | ( D ` x ) = K } ) = { x e. V | -. ( D ` x ) = K }
8	4, 6, 7	3eqtri 2648	. . . 4 |- B = { x e. V | -. ( D ` x ) = K }
9	3, 8	elrab2 3366	. . 3 |- ( Y e. B <-> ( Y e. V /\ -. ( D ` Y ) = K ) )
10		fveq2 6191	. . . . . . 7 |- ( x = X -> ( D ` x ) = ( D ` X ) )
11	10	eqeq1d 2624	. . . . . 6 |- ( x = X -> ( ( D ` x ) = K <-> ( D ` X ) = K ) )
12	11, 5	elrab2 3366	. . . . 5 |- ( X e. A <-> ( X e. V /\ ( D ` X ) = K ) )
13		eqeq2 2633	. . . . . . 7 |- ( ( D ` X ) = K -> ( ( D ` Y ) = ( D ` X ) <-> ( D ` Y ) = K ) )
14	13	notbid 308	. . . . . 6 |- ( ( D ` X ) = K -> ( -. ( D ` Y ) = ( D ` X ) <-> -. ( D ` Y ) = K ) )
15		neqne 2802	. . . . . . 7 |- ( -. ( D ` Y ) = ( D ` X ) -> ( D ` Y ) =/= ( D ` X ) )
16	15	necomd 2849	. . . . . 6 |- ( -. ( D ` Y ) = ( D ` X ) -> ( D ` X ) =/= ( D ` Y ) )
17	14, 16	syl6bir 244	. . . . 5 |- ( ( D ` X ) = K -> ( -. ( D ` Y ) = K -> ( D ` X ) =/= ( D ` Y ) ) )
18	12, 17	simplbiim 659	. . . 4 |- ( X e. A -> ( -. ( D ` Y ) = K -> ( D ` X ) =/= ( D ` Y ) ) )
19	18	com12 32	. . 3 |- ( -. ( D ` Y ) = K -> ( X e. A -> ( D ` X ) =/= ( D ` Y ) ) )
20	9, 19	simplbiim 659	. 2 |- ( Y e. B -> ( X e. A -> ( D ` X ) =/= ( D ` Y ) ) )
21	20	impcom 446	1 |- ( ( X e. A /\ Y e. B ) -> ( D ` X ) =/= ( D ` Y ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   \ cdif 3571   ` cfv 5888  Vtxcvtx 25874  VtxDegcvtxdg 26361

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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

This theorem is referenced by:  frgrwopreglem4  27179  frgrwopreglem5lem  27184