Theorem ivthinclemdisj 15363

Description: Lemma for ivthinc 15366. 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 5639	. . . . . . . 8 |- ( x = z -> ( F ` x ) = ( F ` z ) )
2	1	eleq1d 2300	. . . . . . 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 2605	. . . . . . . 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 5639	. . . . . . . . . . . 12 |- ( w = z -> ( F ` w ) = ( F ` z ) )
7	6	breq1d 4098	. . . . . . . . . . 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 2965	. . . . . . . . . 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 2915	. . . . . 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 8298	. . . . 5 |- ( ( ph /\ z e. L ) -> -. U < ( F ` z ) )
18	17	intnand 938	. . . 4 |- ( ( ph /\ z e. L ) -> -. ( z e. ( A [,] B ) /\ U < ( F ` z ) ) )
19	6	breq2d 4100	. . . . 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 2965	. . . 4 |- ( z e. R <-> ( z e. ( A [,] B ) /\ U < ( F ` z ) ) )
22	18, 21	sylnibr 683	. . 3 |- ( ( ph /\ z e. L ) -> -. z e. R )
23	22	ralrimiva 2605	. 2 |- ( ph -> A. z e. L -. z e. R )
24		disj 3543	. 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 1397   e. wcel 2202   A.wral 2510   {crab 2514   i^i cin 3199   C_ wss 3200   (/)c0 3494   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   CCcc 8029   RRcr 8030   < clt 8213   [,]cicc 10125   -cn->ccncf 15293

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-pre-ltirr 8143  ax-pre-lttrn 8145

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-iota 5286  df-fv 5334  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219

This theorem is referenced by:  ivthinclemex  15365