Theorem dedekindicclemub 14863

Description: Lemma for dedekindicc 14869. 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 2257	. . . 4 |- ( r = a -> ( r e. U <-> a e. U ) )
3	2	cbvrexv 2730	. . 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 14862	. . 3 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> A. y e. L y < a )
27		brralrspcev 4091	. . 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 2619	1 |- ( ph -> E. x e. ( A [,] B ) A. y e. L y < x )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   i^i cin 3156   C_ wss 3157   (/)c0 3450   class class class wbr 4033  (class class class)co 5922   RRcr 7878   < clt 8061   [,]cicc 9966

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-icc 9970

This theorem is referenced by: (None)