Theorem dedekindeulemuub 14499

Description: Lemma for dedekindeu 14505. 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 2252	. . . . 5 |- ( r = A -> ( r e. U <-> A e. U ) )
3		breq2 4022	. . . . . 6 |- ( r = A -> ( q < r <-> q < A ) )
4	3	rexbidv 2491	. . . . 5 |- ( r = A -> ( E. q e. U q < r <-> E. q e. U q < A ) )
5	2, 4	bibi12d 235	. . . 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 3171	. . . 4 |- ( ph -> A e. RR )
9	5, 6, 8	rspcdva 2861	. . 3 |- ( ph -> ( A e. U <-> E. q e. U q < A ) )
10	1, 9	mpbid 147	. 2 |- ( ph -> E. q e. U q < A )
11		dedekindeu.lss	. . . . . 6 |- ( ph -> L C_ RR )
12	11	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> L C_ RR )
13		simpr 110	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> z e. L )
14	12, 13	sseldd 3171	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> z e. RR )
15	7	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> U C_ RR )
16		simplrl 535	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> q e. U )
17	15, 16	sseldd 3171	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> q e. RR )
18	8	ad2antrr 488	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> A e. RR )
19		breq1 4021	. . . . . . . . . 10 |- ( a = q -> ( a < z <-> q < z ) )
20	19	rspcev 2856	. . . . . . . . 9 |- ( ( q e. U /\ q < z ) -> E. a e. U a < z )
21	16, 20	sylan 283	. . . . . . . 8 |- ( ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) /\ q < z ) -> E. a e. U a < z )
22	19	cbvrexv 2719	. . . . . . . 8 |- ( E. a e. U a < z <-> E. q e. U q < z )
23	21, 22	sylib 122	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) /\ q < z ) -> E. q e. U q < z )
24		eleq1 2252	. . . . . . . . 9 |- ( r = z -> ( r e. U <-> z e. U ) )
25		breq2 4022	. . . . . . . . . 10 |- ( r = z -> ( q < r <-> q < z ) )
26	25	rexbidv 2491	. . . . . . . . 9 |- ( r = z -> ( E. q e. U q < r <-> E. q e. U q < z ) )
27	24, 26	bibi12d 235	. . . . . . . 8 |- ( r = z -> ( ( r e. U <-> E. q e. U q < r ) <-> ( z e. U <-> E. q e. U q < z ) ) )
28	6	ad3antrrr 492	. . . . . . . 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 276	. . . . . . . 8 |- ( ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) /\ q < z ) -> z e. RR )
30	27, 28, 29	rspcdva 2861	. . . . . . 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 167	. . . . . 6 |- ( ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) /\ q < z ) -> z e. U )
32		simplll 533	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) /\ q < z ) -> ph )
33	13	adantr 276	. . . . . . 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 3486	. . . . . . . . 9 |- ( ( L i^i U ) = (/) <-> A. z e. L -. z e. U )
36	34, 35	sylib 122	. . . . . . . 8 |- ( ph -> A. z e. L -. z e. U )
37	36	r19.21bi 2578	. . . . . . 7 |- ( ( ph /\ z e. L ) -> -. z e. U )
38	32, 33, 37	syl2anc 411	. . . . . 6 |- ( ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) /\ q < z ) -> -. z e. U )
39	31, 38	pm2.65da 662	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> -. q < z )
40	14, 17, 39	nltled 8098	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> z <_ q )
41		simplrr 536	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> q < A )
42	14, 17, 18, 40, 41	lelttrd 8102	. . 3 |- ( ( ( ph /\ ( q e. U /\ q < A ) ) /\ z e. L ) -> z < A )
43	42	ralrimiva 2563	. 2 |- ( ( ph /\ ( q e. U /\ q < A ) ) -> A. z e. L z < A )
44	10, 43	rexlimddv 2612	1 |- ( ph -> A. z e. L z < A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2160   A.wral 2468   E.wrex 2469   i^i cin 3143   C_ wss 3144   (/)c0 3437   class class class wbr 4018   RRcr 7830   < clt 8012

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7922  ax-resscn 7923  ax-pre-ltwlin 7944

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4647  df-cnv 4649  df-pnf 8014  df-mnf 8015  df-xr 8016  df-ltxr 8017  df-le 8018

This theorem is referenced by:  dedekindeulemub  14500  dedekindeulemloc  14501