Theorem ivthinclemdisj 15631

Description: Lemma for ivthinc 15634. 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 5675	. . . . . . . 8 |- ( x = z -> ( F ` x ) = ( F ` z ) )
2	1	eleq1d 2303	. . . . . . 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 2617	. . . . . . . 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 5675	. . . . . . . . . . . 12 |- ( w = z -> ( F ` w ) = ( F ` z ) )
7	6	breq1d 4124	. . . . . . . . . . 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 2979	. . . . . . . . . 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 2928	. . . . . 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 8409	. . . . 5 |- ( ( ph /\ z e. L ) -> -. U < ( F ` z ) )
18	17	intnand 939	. . . 4 |- ( ( ph /\ z e. L ) -> -. ( z e. ( A [,] B ) /\ U < ( F ` z ) ) )
19	6	breq2d 4126	. . . . 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 2979	. . . 4 |- ( z e. R <-> ( z e. ( A [,] B ) /\ U < ( F ` z ) ) )
22	18, 21	sylnibr 684	. . 3 |- ( ( ph /\ z e. L ) -> -. z e. R )
23	22	ralrimiva 2617	. 2 |- ( ph -> A. z e. L -. z e. R )
24		disj 3561	. 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 1398   e. wcel 2205   A.wral 2522   {crab 2526   i^i cin 3213   C_ wss 3214   (/)c0 3512   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   < clt 8324   [,]cicc 10243   -cn->ccncf 15561

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-pre-ltirr 8255  ax-pre-lttrn 8257

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-cnv 4762  df-iota 5317  df-fv 5365  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330

This theorem is referenced by:  ivthinclemex  15633