Theorem dedekindicclemub 14190

Description: Lemma for dedekindicc 14196. 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 2238	. . . 4 |- ( r = a -> ( r e. U <-> a e. U ) )
3	2	cbvrexv 2706	. . 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 529	. . 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 531	. . . 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 14189	. . 3 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> A. y e. L y < a )
27		brralrspcev 4063	. . 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 2599	1 |- ( ph -> E. x e. ( A [,] B ) A. y e. L y < x )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   i^i cin 3130   C_ wss 3131   (/)c0 3424   class class class wbr 4005  (class class class)co 5877   RRcr 7812   < clt 7994   [,]cicc 9893

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-icc 9897

This theorem is referenced by: (None)