Theorem axcontlem3 23384

Description: Lemma for axcont 23394. 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 1031	. 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 995	. . . . . 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 568	. . . . 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 1032	. . . . . 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 465	. . . . 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 4406	. . . . . 6 |- ( x = U -> ( x Btwn <. Z , y >. <-> U Btwn <. Z , y >. ) )
7		opeq2 4171	. . . . . . 7 |- ( y = p -> <. Z , y >. = <. Z , p >. )
8	7	breq2d 4415	. . . . . 6 |- ( y = p -> ( U Btwn <. Z , y >. <-> U Btwn <. Z , p >. ) )
9	6, 8	rspc2v 3186	. . . . 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 60	. . . 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 392	. . 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 2830	. 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 3492	. . 3 |- ( B C_ D <-> B C_ { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) } )
15		ssrab 3541	. . 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 249	. 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 664	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 368   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2648   A.wral 2799   {crab 2803   C_ wss 3439   <.cop 3994   class class class wbr 4403   ` cfv 5529   NNcn 10436   EEcee 23306   Btwn cbtwn 23307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404

This theorem is referenced by:  axcontlem9  23390  axcontlem10  23391