Theorem dedekindeulemub 13236

Description: Lemma for dedekindeu 13241. 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 2227	. . . 4 |- ( r = a -> ( r e. U <-> a e. U ) )
3	2	cbvrexv 2693	. . 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 13235	. . 3 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> A. y e. L y < a )
23		brralrspcev 4040	. . 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 2588	1 |- ( ph -> E. x e. RR A. y e. L y < x )

Syntax hints:   -> 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:  dedekindeulemlub  13238