Theorem ivthinclemdisj 13018

Description: Lemma for ivthinc 13021. The lower and upper cuts are disjoint. (Contributed by Jim Kingdon, 18-Feb-2024.)

Hypotheses

Ref	Expression
ivth.1	|- ( ph -> A e. RR )
ivth.2	|- ( ph -> B e. RR )
ivth.3	|- ( ph -> U e. RR )
ivth.4	|- ( ph -> A < B )
ivth.5	|- ( ph -> ( A [,] B ) C_ D )
ivth.7	|- ( ph -> F e. ( D -cn-> CC ) )
ivth.8	|- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
ivth.9	|- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
ivthinc.i	|- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )
ivthinclem.l	|- L = { w e. ( A [,] B ) | ( F ` w ) < U }
ivthinclem.r	|- R = { w e. ( A [,] B ) | U < ( F ` w ) }

Assertion

Ref	Expression
ivthinclemdisj	|- ( ph -> ( L i^i R ) = (/) )

Distinct variable groups:   w, A   x, A   w, B   x, B   w, F   x, F   w, U   ph, x

Allowed substitution hints:   ph( y, w)   A( y)   B( y)   D( x, y, w)   R( x, y, w)   U( x, y)   F( y)   L( x, y, w)

Proof of Theorem ivthinclemdisj

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5468	. . . . . . . 8 |- ( x = z -> ( F ` x ) = ( F ` z ) )
2	1	eleq1d 2226	. . . . . . 7 |- ( x = z -> ( ( F ` x ) e. RR <-> ( F ` z ) e. RR ) )
3		ivth.8	. . . . . . . . 9 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
4	3	ralrimiva 2530	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
5	4	adantr 274	. . . . . . 7 |- ( ( ph /\ z e. L ) -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
6		fveq2 5468	. . . . . . . . . . . 12 |- ( w = z -> ( F ` w ) = ( F ` z ) )
7	6	breq1d 3975	. . . . . . . . . . 11 |- ( w = z -> ( ( F ` w ) < U <-> ( F ` z ) < U ) )
8		ivthinclem.l	. . . . . . . . . . 11 |- L = { w e. ( A [,] B ) | ( F ` w ) < U }
9	7, 8	elrab2 2871	. . . . . . . . . 10 |- ( z e. L <-> ( z e. ( A [,] B ) /\ ( F ` z ) < U ) )
10	9	biimpi 119	. . . . . . . . 9 |- ( z e. L -> ( z e. ( A [,] B ) /\ ( F ` z ) < U ) )
11	10	adantl 275	. . . . . . . 8 |- ( ( ph /\ z e. L ) -> ( z e. ( A [,] B ) /\ ( F ` z ) < U ) )
12	11	simpld 111	. . . . . . 7 |- ( ( ph /\ z e. L ) -> z e. ( A [,] B ) )
13	2, 5, 12	rspcdva 2821	. . . . . 6 |- ( ( ph /\ z e. L ) -> ( F ` z ) e. RR )
14		ivth.3	. . . . . . 7 |- ( ph -> U e. RR )
15	14	adantr 274	. . . . . 6 |- ( ( ph /\ z e. L ) -> U e. RR )
16	11	simprd 113	. . . . . 6 |- ( ( ph /\ z e. L ) -> ( F ` z ) < U )
17	13, 15, 16	ltnsymd 7995	. . . . 5 |- ( ( ph /\ z e. L ) -> -. U < ( F ` z ) )
18	17	intnand 917	. . . 4 |- ( ( ph /\ z e. L ) -> -. ( z e. ( A [,] B ) /\ U < ( F ` z ) ) )
19	6	breq2d 3977	. . . . 5 |- ( w = z -> ( U < ( F ` w ) <-> U < ( F ` z ) ) )
20		ivthinclem.r	. . . . 5 |- R = { w e. ( A [,] B ) | U < ( F ` w ) }
21	19, 20	elrab2 2871	. . . 4 |- ( z e. R <-> ( z e. ( A [,] B ) /\ U < ( F ` z ) ) )
22	18, 21	sylnibr 667	. . 3 |- ( ( ph /\ z e. L ) -> -. z e. R )
23	22	ralrimiva 2530	. 2 |- ( ph -> A. z e. L -. z e. R )
24		disj 3442	. 2 |- ( ( L i^i R ) = (/) <-> A. z e. L -. z e. R )
25	23, 24	sylibr 133	1 |- ( ph -> ( L i^i R ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   A.wral 2435   {crab 2439   i^i cin 3101   C_ wss 3102   (/)c0 3394   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   CCcc 7730   RRcr 7731   < clt 7912   [,]cicc 9795   -cn->ccncf 12957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-pre-ltirr 7844  ax-pre-lttrn 7846

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4592  df-cnv 4594  df-iota 5135  df-fv 5178  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918

This theorem is referenced by:  ivthinclemex  13020