Theorem dedekindicclemub 13399

Description: Lemma for dedekindicc 13405. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)

Hypotheses

Ref	Expression
dedekindicc.a	|- ( ph -> A e. RR )
dedekindicc.b	|- ( ph -> B e. RR )
dedekindicc.lss	|- ( ph -> L C_ ( A [,] B ) )
dedekindicc.uss	|- ( ph -> U C_ ( A [,] B ) )
dedekindicc.lm	|- ( ph -> E. q e. ( A [,] B ) q e. L )
dedekindicc.um	|- ( ph -> E. r e. ( A [,] B ) r e. U )
dedekindicc.lr	|- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )
dedekindicc.ur	|- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )
dedekindicc.disj	|- ( ph -> ( L i^i U ) = (/) )
dedekindicc.loc	|- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )

Assertion

Ref	Expression
dedekindicclemub	|- ( ph -> E. x e. ( A [,] B ) A. y e. L y < x )

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

Allowed substitution hints:   ph( x, r)   U( x)   L( r)

Proof of Theorem dedekindicclemub

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dedekindicc.um	. . 3 |- ( ph -> E. r e. ( A [,] B ) r e. U )
2		eleq1w 2231	. . . 4 |- ( r = a -> ( r e. U <-> a e. U ) )
3	2	cbvrexv 2697	. . 3 |- ( E. r e. ( A [,] B ) r e. U <-> E. a e. ( A [,] B ) a e. U )
4	1, 3	sylib 121	. 2 |- ( ph -> E. a e. ( A [,] B ) a e. U )
5		simprl 526	. . 3 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> a e. ( A [,] B ) )
6		dedekindicc.a	. . . . 5 |- ( ph -> A e. RR )
7	6	adantr 274	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> A e. RR )
8		dedekindicc.b	. . . . 5 |- ( ph -> B e. RR )
9	8	adantr 274	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> B e. RR )
10		dedekindicc.lss	. . . . 5 |- ( ph -> L C_ ( A [,] B ) )
11	10	adantr 274	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> L C_ ( A [,] B ) )
12		dedekindicc.uss	. . . . 5 |- ( ph -> U C_ ( A [,] B ) )
13	12	adantr 274	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> U C_ ( A [,] B ) )
14		dedekindicc.lm	. . . . 5 |- ( ph -> E. q e. ( A [,] B ) q e. L )
15	14	adantr 274	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> E. q e. ( A [,] B ) q e. L )
16	1	adantr 274	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> E. r e. ( A [,] B ) r e. U )
17		dedekindicc.lr	. . . . 5 |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )
18	17	adantr 274	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )
19		dedekindicc.ur	. . . . 5 |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )
20	19	adantr 274	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )
21		dedekindicc.disj	. . . . 5 |- ( ph -> ( L i^i U ) = (/) )
22	21	adantr 274	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> ( L i^i U ) = (/) )
23		dedekindicc.loc	. . . . 5 |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )
24	23	adantr 274	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )
25		simprr 527	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> a e. U )
26	7, 9, 11, 13, 15, 16, 18, 20, 22, 24, 25	dedekindicclemuub 13398	. . 3 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> A. y e. L y < a )
27		brralrspcev 4047	. . 3 |- ( ( a e. ( A [,] B ) /\ A. y e. L y < a ) -> E. x e. ( A [,] B ) A. y e. L y < x )
28	5, 26, 27	syl2anc 409	. 2 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> E. x e. ( A [,] B ) A. y e. L y < x )
29	4, 28	rexlimddv 2592	1 |- ( ph -> E. x e. ( A [,] B ) A. y e. L y < x )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   i^i cin 3120   C_ wss 3121   (/)c0 3414   class class class wbr 3989  (class class class)co 5853   RRcr 7773   < clt 7954   [,]cicc 9848

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-icc 9852

This theorem is referenced by: (None)