Theorem axcontlem3 24474

Description: Lemma for axcont 24484. 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 1037	. 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 1001	. . . . . 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 566	. . . . 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 1038	. . . . . 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 463	. . . . 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 4442	. . . . . 6 |- ( x = U -> ( x Btwn <. Z , y >. <-> U Btwn <. Z , y >. ) )
7		opeq2 4204	. . . . . . 7 |- ( y = p -> <. Z , y >. = <. Z , p >. )
8	7	breq2d 4451	. . . . . 6 |- ( y = p -> ( U Btwn <. Z , y >. <-> U Btwn <. Z , p >. ) )
9	6, 8	rspc2v 3216	. . . . 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 390	. . 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 2868	. 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 3514	. . 3 |- ( B C_ D <-> B C_ { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) } )
15		ssrab 3564	. . 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 662	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 366   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   {crab 2808   C_ wss 3461   <.cop 4022   class class class wbr 4439   ` cfv 5570   NNcn 10531   EEcee 24396   Btwn cbtwn 24397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440

This theorem is referenced by:  axcontlem9  24480  axcontlem10  24481