Theorem dedekindeulemub 14797

Description: Lemma for dedekindeu 14802. 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 2254	. . . 4 |- ( r = a -> ( r e. U <-> a e. U ) )
3	2	cbvrexv 2727	. . 3 |- ( E. r e. RR r e. U <-> E. a e. RR a e. U )
4	1, 3	sylib 122	. 2 |- ( ph -> E. a e. RR a e. U )
5		simprl 529	. . 3 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> a e. RR )
6		dedekindeu.lss	. . . . 5 |- ( ph -> L C_ RR )
7	6	adantr 276	. . . 4 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> L C_ RR )
8		dedekindeu.uss	. . . . 5 |- ( ph -> U C_ RR )
9	8	adantr 276	. . . 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 276	. . . 4 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> E. q e. RR q e. L )
12	1	adantr 276	. . . 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 276	. . . 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 276	. . . 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 276	. . . 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 276	. . . 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 531	. . . 4 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> a e. U )
22	7, 9, 11, 12, 14, 16, 18, 20, 21	dedekindeulemuub 14796	. . 3 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> A. y e. L y < a )
23		brralrspcev 4088	. . 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 411	. 2 |- ( ( ph /\ ( a e. RR /\ a e. U ) ) -> E. x e. RR A. y e. L y < x )
25	4, 24	rexlimddv 2616	1 |- ( ph -> E. x e. RR A. y e. L y < x )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   i^i cin 3153   C_ wss 3154   (/)c0 3447   class class class wbr 4030   RRcr 7873   < clt 8056

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-pre-ltwlin 7987

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-xp 4666  df-cnv 4668  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062

This theorem is referenced by:  dedekindeulemlub  14799