Theorem dedekindeulemub 12804

Description: Lemma for dedekindeu 12809. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-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 ) ) )

Assertion

Ref	Expression
dedekindeulemub	|- ( ph -> E. x e. RR A. y e. L y < x )

Distinct variable groups:   L, q, r, y   x, L, r, y   U, q, r, y   ph, q, r, y

Allowed substitution hints:   ph( x)   U( x)

Proof of Theorem dedekindeulemub

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dedekindeu.um	. . 3 |- ( ph -> E. r e. RR r e. U )
2		eleq1w 2201	. . . 4 |- ( r = a -> ( r e. U <-> a e. U ) )
3	2	cbvrexv 2658	. . 3 |- ( E. r e. RR r e. U <-> E. a e. RR a e. U )
4	1, 3	sylib 121	. 2 |- ( ph -> E. a e. RR a e. U )
5		simprl 521	. . 3 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> a e. RR )
6		dedekindeu.lss	. . . . 5 |- ( ph -> L C_ RR )
7	6	adantr 274	. . . 4 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> L C_ RR )
8		dedekindeu.uss	. . . . 5 |- ( ph -> U C_ RR )
9	8	adantr 274	. . . 4 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> U C_ RR )
10		dedekindeu.lm	. . . . 5 |- ( ph -> E. q e. RR q e. L )
11	10	adantr 274	. . . 4 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> E. q e. RR q e. L )
12	1	adantr 274	. . . 4 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> E. r e. RR r e. U )
13		dedekindeu.lr	. . . . 5 |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )
14	13	adantr 274	. . . 4 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )
15		dedekindeu.ur	. . . . 5 |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )
16	15	adantr 274	. . . 4 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )
17		dedekindeu.disj	. . . . 5 |- ( ph -> ( L i^i U ) = (/) )
18	17	adantr 274	. . . 4 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> ( L i^i U ) = (/) )
19		dedekindeu.loc	. . . . 5 |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )
20	19	adantr 274	. . . 4 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )
21		simprr 522	. . . 4 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> a e. U )
22	7, 9, 11, 12, 14, 16, 18, 20, 21	dedekindeulemuub 12803	. . 3 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> A. y e. L y < a )
23		brralrspcev 3994	. . 3 |- ( ( a e. RR /\ A. y e. L y < a ) -> E. x e. RR A. y e. L y < x )
24	5, 22, 23	syl2anc 409	. 2 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> E. x e. RR A. y e. L y < x )
25	4, 24	rexlimddv 2557	1 |- ( ph -> E. x e. RR A. y e. L y < x )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1332   e. wcel 1481   A.wral 2417   E.wrex 2418   i^i cin 3075   C_ wss 3076   (/)c0 3368   class class class wbr 3937   RRcr 7643   < clt 7824

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-pre-ltwlin 7757

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830

This theorem is referenced by:  dedekindeulemlub  12806