Theorem axcontlem3 25846

Description: Lemma for axcont 25856. Given the separation assumption, B is a subset of D. (Contributed by Scott Fenton, 18-Jun-2013.)

Hypothesis

Ref	Expression
axcontlem3.1	|- D = { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) }

Assertion

Ref	Expression
axcontlem3	|- ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ ( Z e. ( EE ` N ) /\ U e. A /\ Z =/= U ) ) -> B C_ D )

Distinct variable groups:   A, p, x   B, p, x, y   N, p, x, y   U, p, x, y   Z, p, x, y

Allowed substitution hints:   A( y)   D( x, y, p)

Proof of Theorem axcontlem3

Step	Hyp	Ref	Expression
1		simplr2 1104	. 2 |- ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ ( Z e. ( EE ` N ) /\ U e. A /\ Z =/= U ) ) -> B C_ ( EE ` N ) )
2		simpr2 1068	. . . . . 6 |- ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ ( Z e. ( EE ` N ) /\ U e. A /\ Z =/= U ) ) -> U e. A )
3	2	anim1i 592	. . . . 5 |- ( ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ ( Z e. ( EE ` N ) /\ U e. A /\ Z =/= U ) ) /\ p e. B ) -> ( U e. A /\ p e. B ) )
4		simplr3 1105	. . . . . 6 |- ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ ( Z e. ( EE ` N ) /\ U e. A /\ Z =/= U ) ) -> A. x e. A A. y e. B x Btwn <. Z , y >. )
5	4	adantr 481	. . . . 5 |- ( ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ ( Z e. ( EE ` N ) /\ U e. A /\ Z =/= U ) ) /\ p e. B ) -> A. x e. A A. y e. B x Btwn <. Z , y >. )
6		breq1 4656	. . . . . 6 |- ( x = U -> ( x Btwn <. Z , y >. <-> U Btwn <. Z , y >. ) )
7		opeq2 4403	. . . . . . 7 |- ( y = p -> <. Z , y >. = <. Z , p >. )
8	7	breq2d 4665	. . . . . 6 |- ( y = p -> ( U Btwn <. Z , y >. <-> U Btwn <. Z , p >. ) )
9	6, 8	rspc2v 3322	. . . . 5 |- ( ( U e. A /\ p e. B ) -> ( A. x e. A A. y e. B x Btwn <. Z , y >. -> U Btwn <. Z , p >. ) )
10	3, 5, 9	sylc 65	. . . 4 |- ( ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ ( Z e. ( EE ` N ) /\ U e. A /\ Z =/= U ) ) /\ p e. B ) -> U Btwn <. Z , p >. )
11	10	orcd 407	. . 3 |- ( ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ ( Z e. ( EE ` N ) /\ U e. A /\ Z =/= U ) ) /\ p e. B ) -> ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) )
12	11	ralrimiva 2966	. 2 |- ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ ( Z e. ( EE ` N ) /\ U e. A /\ Z =/= U ) ) -> A. p e. B ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) )
13		axcontlem3.1	. . . 4 |- D = { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) }
14	13	sseq2i 3630	. . 3 |- ( B C_ D <-> B C_ { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) } )
15		ssrab 3680	. . 3 |- ( B C_ { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) } <-> ( B C_ ( EE ` N ) /\ A. p e. B ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) ) )
16	14, 15	bitri 264	. 2 |- ( B C_ D <-> ( B C_ ( EE ` N ) /\ A. p e. B ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) ) )
17	1, 12, 16	sylanbrc 698	1 |- ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ ( Z e. ( EE ` N ) /\ U e. A /\ Z =/= U ) ) -> B C_ D )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   C_ wss 3574   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769

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-ral 2917  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-br 4654

This theorem is referenced by:  axcontlem9  25852  axcontlem10  25853