Theorem dedekindicclemuub 13398

Description: Lemma for dedekindicc 13405. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 15-Feb-2024.)

Hypotheses

Ref	Expression
dedekindicc.a	|- ( ph -> A e. RR )
dedekindicc.b	|- ( ph -> B e. RR )
dedekindicc.lss	|- ( ph -> L C_ ( A [,] B ) )
dedekindicc.uss	|- ( ph -> U C_ ( A [,] B ) )
dedekindicc.lm	|- ( ph -> E. q e. ( A [,] B ) q e. L )
dedekindicc.um	|- ( ph -> E. r e. ( A [,] B ) r e. U )
dedekindicc.lr	|- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )
dedekindicc.ur	|- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )
dedekindicc.disj	|- ( ph -> ( L i^i U ) = (/) )
dedekindicc.loc	|- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )
dedekindicclemuub.u	|- ( ph -> C e. U )

Assertion

Ref	Expression
dedekindicclemuub	|- ( ph -> A. z e. L z < C )

Distinct variable groups:   A, r   B, r   C, q, r, z   L, q, z   U, q, z, r   ph, q, z

Allowed substitution hints:   ph( r)   A( z, q)   B( z, q)   L( r)

Proof of Theorem dedekindicclemuub

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dedekindicclemuub.u	. . 3 |- ( ph -> C e. U )
2		eleq1 2233	. . . . 5 |- ( r = C -> ( r e. U <-> C e. U ) )
3		breq2 3993	. . . . . 6 |- ( r = C -> ( q < r <-> q < C ) )
4	3	rexbidv 2471	. . . . 5 |- ( r = C -> ( E. q e. U q < r <-> E. q e. U q < C ) )
5	2, 4	bibi12d 234	. . . 4 |- ( r = C -> ( ( r e. U <-> E. q e. U q < r ) <-> ( C e. U <-> E. q e. U q < C ) ) )
6		dedekindicc.ur	. . . 4 |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )
7		dedekindicc.uss	. . . . 5 |- ( ph -> U C_ ( A [,] B ) )
8	7, 1	sseldd 3148	. . . 4 |- ( ph -> C e. ( A [,] B ) )
9	5, 6, 8	rspcdva 2839	. . 3 |- ( ph -> ( C e. U <-> E. q e. U q < C ) )
10	1, 9	mpbid 146	. 2 |- ( ph -> E. q e. U q < C )
11		dedekindicc.a	. . . . . . 7 |- ( ph -> A e. RR )
12		dedekindicc.b	. . . . . . 7 |- ( ph -> B e. RR )
13		iccssre 9912	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
14	11, 12, 13	syl2anc 409	. . . . . 6 |- ( ph -> ( A [,] B ) C_ RR )
15	14	ad2antrr 485	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> ( A [,] B ) C_ RR )
16		dedekindicc.lss	. . . . . . 7 |- ( ph -> L C_ ( A [,] B ) )
17	16	ad2antrr 485	. . . . . 6 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> L C_ ( A [,] B ) )
18		simpr 109	. . . . . 6 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z e. L )
19	17, 18	sseldd 3148	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z e. ( A [,] B ) )
20	15, 19	sseldd 3148	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z e. RR )
21	7	ad2antrr 485	. . . . . 6 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> U C_ ( A [,] B ) )
22		simplrl 530	. . . . . 6 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q e. U )
23	21, 22	sseldd 3148	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q e. ( A [,] B ) )
24	15, 23	sseldd 3148	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q e. RR )
25	14, 8	sseldd 3148	. . . . 5 |- ( ph -> C e. RR )
26	25	ad2antrr 485	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> C e. RR )
27		breq1 3992	. . . . . . . . . 10 |- ( a = q -> ( a < z <-> q < z ) )
28	27	rspcev 2834	. . . . . . . . 9 |- ( ( q e. U /\ q < z ) -> E. a e. U a < z )
29	22, 28	sylan 281	. . . . . . . 8 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> E. a e. U a < z )
30	27	cbvrexv 2697	. . . . . . . 8 |- ( E. a e. U a < z <-> E. q e. U q < z )
31	29, 30	sylib 121	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> E. q e. U q < z )
32		eleq1 2233	. . . . . . . . 9 |- ( r = z -> ( r e. U <-> z e. U ) )
33		breq2 3993	. . . . . . . . . 10 |- ( r = z -> ( q < r <-> q < z ) )
34	33	rexbidv 2471	. . . . . . . . 9 |- ( r = z -> ( E. q e. U q < r <-> E. q e. U q < z ) )
35	32, 34	bibi12d 234	. . . . . . . 8 |- ( r = z -> ( ( r e. U <-> E. q e. U q < r ) <-> ( z e. U <-> E. q e. U q < z ) ) )
36	6	ad3antrrr 489	. . . . . . . 8 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )
37	19	adantr 274	. . . . . . . 8 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> z e. ( A [,] B ) )
38	35, 36, 37	rspcdva 2839	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> ( z e. U <-> E. q e. U q < z ) )
39	31, 38	mpbird 166	. . . . . 6 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> z e. U )
40		simplll 528	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> ph )
41	18	adantr 274	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> z e. L )
42		dedekindicc.disj	. . . . . . . . 9 |- ( ph -> ( L i^i U ) = (/) )
43		disj 3463	. . . . . . . . 9 |- ( ( L i^i U ) = (/) <-> A. z e. L -. z e. U )
44	42, 43	sylib 121	. . . . . . . 8 |- ( ph -> A. z e. L -. z e. U )
45	44	r19.21bi 2558	. . . . . . 7 |- ( ( ph /\ z e. L ) -> -. z e. U )
46	40, 41, 45	syl2anc 409	. . . . . 6 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> -. z e. U )
47	39, 46	pm2.65da 656	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> -. q < z )
48	20, 24, 47	nltled 8040	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z <_ q )
49		simplrr 531	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q < C )
50	20, 24, 26, 48, 49	lelttrd 8044	. . 3 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z < C )
51	50	ralrimiva 2543	. 2 |- ( ( ph /\ ( q e. U /\ q < C ) ) -> A. z e. L z < C )
52	10, 51	rexlimddv 2592	1 |- ( ph -> A. z e. L z < C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   i^i cin 3120   C_ wss 3121   (/)c0 3414   class class class wbr 3989  (class class class)co 5853   RRcr 7773   < clt 7954   [,]cicc 9848

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-icc 9852

This theorem is referenced by:  dedekindicclemub  13399  dedekindicclemloc  13400