Theorem dedekindicclemub 15418

Description: Lemma for dedekindicc 15424. 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 2292	. . . 4 |- ( r = a -> ( r e. U <-> a e. U ) )
3	2	cbvrexv 2769	. . 3 |- ( E. r e. ( A [,] B ) r e. U <-> E. a e. ( A [,] B ) a e. U )
4	1, 3	sylib 122	. 2 |- ( ph -> E. a e. ( A [,] B ) a e. U )
5		simprl 531	. . 3 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> a e. ( A [,] B ) )
6		dedekindicc.a	. . . . 5 |- ( ph -> A e. RR )
7	6	adantr 276	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> A e. RR )
8		dedekindicc.b	. . . . 5 |- ( ph -> B e. RR )
9	8	adantr 276	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> B e. RR )
10		dedekindicc.lss	. . . . 5 |- ( ph -> L C_ ( A [,] B ) )
11	10	adantr 276	. . . 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 276	. . . 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 276	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> E. q e. ( A [,] B ) q e. L )
16	1	adantr 276	. . . 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 276	. . . 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 276	. . . 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 276	. . . 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 276	. . . 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 533	. . . 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 15417	. . 3 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> A. y e. L y < a )
27		brralrspcev 4152	. . 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 411	. 2 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> E. x e. ( A [,] B ) A. y e. L y < x )
29	4, 28	rexlimddv 2656	1 |- ( ph -> E. x e. ( A [,] B ) A. y e. L y < x )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   i^i cin 3200   C_ wss 3201   (/)c0 3496   class class class wbr 4093  (class class class)co 6028   RRcr 8074   < clt 8257   [,]cicc 10169

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-icc 10173

This theorem is referenced by: (None)