Theorem dedekindicclemuub 12773

Description: Lemma for dedekindicc 12780. 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 2202	. . . . 5 |- ( r = C -> ( r e. U <-> C e. U ) )
3		breq2 3933	. . . . . 6 |- ( r = C -> ( q < r <-> q < C ) )
4	3	rexbidv 2438	. . . . 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 3098	. . . 4 |- ( ph -> C e. ( A [,] B ) )
9	5, 6, 8	rspcdva 2794	. . 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 9738	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
14	11, 12, 13	syl2anc 408	. . . . . 6 |- ( ph -> ( A [,] B ) C_ RR )
15	14	ad2antrr 479	. . . . 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 479	. . . . . 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 3098	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z e. ( A [,] B ) )
20	15, 19	sseldd 3098	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z e. RR )
21	7	ad2antrr 479	. . . . . 6 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> U C_ ( A [,] B ) )
22		simplrl 524	. . . . . 6 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q e. U )
23	21, 22	sseldd 3098	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q e. ( A [,] B ) )
24	15, 23	sseldd 3098	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q e. RR )
25	14, 8	sseldd 3098	. . . . 5 |- ( ph -> C e. RR )
26	25	ad2antrr 479	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> C e. RR )
27		breq1 3932	. . . . . . . . . 10 |- ( a = q -> ( a < z <-> q < z ) )
28	27	rspcev 2789	. . . . . . . . 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 2655	. . . . . . . 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 2202	. . . . . . . . 9 |- ( r = z -> ( r e. U <-> z e. U ) )
33		breq2 3933	. . . . . . . . . 10 |- ( r = z -> ( q < r <-> q < z ) )
34	33	rexbidv 2438	. . . . . . . . 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 483	. . . . . . . 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 2794	. . . . . . 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 522	. . . . . . 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 3411	. . . . . . . . 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 2520	. . . . . . 7 |- ( ( ph /\ z e. L ) -> -. z e. U )
46	40, 41, 45	syl2anc 408	. . . . . 6 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> -. z e. U )
47	39, 46	pm2.65da 650	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> -. q < z )
48	20, 24, 47	nltled 7883	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z <_ q )
49		simplrr 525	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q < C )
50	20, 24, 26, 48, 49	lelttrd 7887	. . 3 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z < C )
51	50	ralrimiva 2505	. 2 |- ( ( ph /\ ( q e. U /\ q < C ) ) -> A. z e. L z < C )
52	10, 51	rexlimddv 2554	1 |- ( ph -> A. z e. L z < C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   = wceq 1331   e. wcel 1480   A.wral 2416   E.wrex 2417   i^i cin 3070   C_ wss 3071   (/)c0 3363   class class class wbr 3929  (class class class)co 5774   RRcr 7619   < clt 7800   [,]cicc 9674

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-icc 9678

This theorem is referenced by:  dedekindicclemub  12774  dedekindicclemloc  12775