Theorem ivthinclemdisj 14794

Description: Lemma for ivthinc 14797. 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 5554	. . . . . . . 8 |- ( x = z -> ( F ` x ) = ( F ` z ) )
2	1	eleq1d 2262	. . . . . . 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 2567	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
5	4	adantr 276	. . . . . . 7 |- ( ( ph /\ z e. L ) -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
6		fveq2 5554	. . . . . . . . . . . 12 |- ( w = z -> ( F ` w ) = ( F ` z ) )
7	6	breq1d 4039	. . . . . . . . . . 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 2919	. . . . . . . . . 10 |- ( z e. L <-> ( z e. ( A [,] B ) /\ ( F ` z ) < U ) )
10	9	biimpi 120	. . . . . . . . 9 |- ( z e. L -> ( z e. ( A [,] B ) /\ ( F ` z ) < U ) )
11	10	adantl 277	. . . . . . . 8 |- ( ( ph /\ z e. L ) -> ( z e. ( A [,] B ) /\ ( F ` z ) < U ) )
12	11	simpld 112	. . . . . . 7 |- ( ( ph /\ z e. L ) -> z e. ( A [,] B ) )
13	2, 5, 12	rspcdva 2869	. . . . . 6 |- ( ( ph /\ z e. L ) -> ( F ` z ) e. RR )
14		ivth.3	. . . . . . 7 |- ( ph -> U e. RR )
15	14	adantr 276	. . . . . 6 |- ( ( ph /\ z e. L ) -> U e. RR )
16	11	simprd 114	. . . . . 6 |- ( ( ph /\ z e. L ) -> ( F ` z ) < U )
17	13, 15, 16	ltnsymd 8139	. . . . 5 |- ( ( ph /\ z e. L ) -> -. U < ( F ` z ) )
18	17	intnand 932	. . . 4 |- ( ( ph /\ z e. L ) -> -. ( z e. ( A [,] B ) /\ U < ( F ` z ) ) )
19	6	breq2d 4041	. . . . 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 2919	. . . 4 |- ( z e. R <-> ( z e. ( A [,] B ) /\ U < ( F ` z ) ) )
22	18, 21	sylnibr 678	. . 3 |- ( ( ph /\ z e. L ) -> -. z e. R )
23	22	ralrimiva 2567	. 2 |- ( ph -> A. z e. L -. z e. R )
24		disj 3495	. 2 |- ( ( L i^i R ) = (/) <-> A. z e. L -. z e. R )
25	23, 24	sylibr 134	1 |- ( ph -> ( L i^i R ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   {crab 2476   i^i cin 3152   C_ wss 3153   (/)c0 3446   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   CCcc 7870   RRcr 7871   < clt 8054   [,]cicc 9957   -cn->ccncf 14725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-pre-ltirr 7984  ax-pre-lttrn 7986

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-cnv 4667  df-iota 5215  df-fv 5262  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060

This theorem is referenced by:  ivthinclemex  14796