Theorem dedekindeulemuub 13235

Description: Lemma for dedekindeu 13241. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-2024.)

Hypotheses

Ref	Expression
dedekindeu.lss	|- ( ph -> L C_ RR )
dedekindeu.uss	|- ( ph -> U C_ RR )
dedekindeu.lm	|- ( ph -> E. q e. RR q e. L )
dedekindeu.um	|- ( ph -> E. r e. RR r e. U )
dedekindeu.lr	|- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )
dedekindeu.ur	|- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )
dedekindeu.disj	|- ( ph -> ( L i^i U ) = (/) )
dedekindeu.loc	|- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )
dedekindeulemuub.u	|- ( ph -> A e. U )

Assertion

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

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

Allowed substitution hints:   ph( r)   L( r)

Proof of Theorem dedekindeulemuub

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dedekindeulemuub.u	. . 3 |- ( ph -> A e. U )
2		eleq1 2229	. . . . 5 |- ( r = A -> ( r e. U <-> A e. U ) )
3		breq2 3986	. . . . . 6 |- ( r = A -> ( q < r <-> q < A ) )
4	3	rexbidv 2467	. . . . 5 |- ( r = A -> ( E. q e. U q < r <-> E. q e. U q < A ) )
5	2, 4	bibi12d 234	. . . 4 |- ( r = A -> ( ( r e. U <-> E. q e. U q < r ) <-> ( A e. U <-> E. q e. U q < A ) ) )
6		dedekindeu.ur	. . . 4 |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )
7		dedekindeu.uss	. . . . 5 |- ( ph -> U C_ RR )
8	7, 1	sseldd 3143	. . . 4 |- ( ph -> A e. RR )
9	5, 6, 8	rspcdva 2835	. . 3 |- ( ph -> ( A e. U <-> E. q e. U q < A ) )
10	1, 9	mpbid 146	. 2 |- ( ph -> E. q e. U q < A )
11		dedekindeu.lss	. . . . . 6 |- ( ph -> L C_ RR )
12	11	ad2antrr 480	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> L C_ RR )
13		simpr 109	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> z e. L )
14	12, 13	sseldd 3143	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> z e. RR )
15	7	ad2antrr 480	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> U C_ RR )
16		simplrl 525	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> q e. U )
17	15, 16	sseldd 3143	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> q e. RR )
18	8	ad2antrr 480	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> A e. RR )
19		breq1 3985	. . . . . . . . . 10 |- ( a = q -> ( a < z <-> q < z ) )
20	19	rspcev 2830	. . . . . . . . 9 |- ( ( q e. U /\ q < z ) -> E. a e. U a < z )
21	16, 20	sylan 281	. . . . . . . 8 |- ( ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) /\ q < z ) -> E. a e. U a < z )
22	19	cbvrexv 2693	. . . . . . . 8 |- ( E. a e. U a < z <-> E. q e. U q < z )
23	21, 22	sylib 121	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) /\ q < z ) -> E. q e. U q < z )
24		eleq1 2229	. . . . . . . . 9 |- ( r = z -> ( r e. U <-> z e. U ) )
25		breq2 3986	. . . . . . . . . 10 |- ( r = z -> ( q < r <-> q < z ) )
26	25	rexbidv 2467	. . . . . . . . 9 |- ( r = z -> ( E. q e. U q < r <-> E. q e. U q < z ) )
27	24, 26	bibi12d 234	. . . . . . . 8 |- ( r = z -> ( ( r e. U <-> E. q e. U q < r ) <-> ( z e. U <-> E. q e. U q < z ) ) )
28	6	ad3antrrr 484	. . . . . . . 8 |- ( ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) /\ q < z ) -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )
29	14	adantr 274	. . . . . . . 8 |- ( ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) /\ q < z ) -> z e. RR )
30	27, 28, 29	rspcdva 2835	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) /\ q < z ) -> ( z e. U <-> E. q e. U q < z ) )
31	23, 30	mpbird 166	. . . . . 6 |- ( ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) /\ q < z ) -> z e. U )
32		simplll 523	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) /\ q < z ) -> ph )
33	13	adantr 274	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) /\ q < z ) -> z e. L )
34		dedekindeu.disj	. . . . . . . . 9 |- ( ph -> ( L i^i U ) = (/) )
35		disj 3457	. . . . . . . . 9 |- ( ( L i^i U ) = (/) <-> A. z e. L -. z e. U )
36	34, 35	sylib 121	. . . . . . . 8 |- ( ph -> A. z e. L -. z e. U )
37	36	r19.21bi 2554	. . . . . . 7 |- ( ( ph /\ z e. L ) -> -. z e. U )
38	32, 33, 37	syl2anc 409	. . . . . 6 |- ( ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) /\ q < z ) -> -. z e. U )
39	31, 38	pm2.65da 651	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> -. q < z )
40	14, 17, 39	nltled 8019	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> z <_ q )
41		simplrr 526	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> q < A )
42	14, 17, 18, 40, 41	lelttrd 8023	. . 3 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> z < A )
43	42	ralrimiva 2539	. 2 |- ( ( ph /\ ( q e. U /\ q < A ) ) -> A. z e. L z < A )
44	10, 43	rexlimddv 2588	1 |- ( ph -> A. z e. L z < A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   i^i cin 3115   C_ wss 3116   (/)c0 3409   class class class wbr 3982   RRcr 7752   < clt 7933

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-pre-ltwlin 7866

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939

This theorem is referenced by:  dedekindeulemub  13236  dedekindeulemloc  13237