Theorem dedekindicclemuub 14862

Description: Lemma for dedekindicc 14869. Any element of the upper cut is an upper bound for the lower cut. (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 ) ) )
dedekindicclemuub.u	|- ( ph -> C e. U )

Assertion

Ref	Expression
dedekindicclemuub	|- ( ph -> A. z e. L z < C )

Distinct variable groups:   A, r   B, r   C, q, r, z   L, q, z   U, q, z, r   ph, q, z

Allowed substitution hints:   ph( r)   A( z, q)   B( z, q)   L( r)

Proof of Theorem dedekindicclemuub

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dedekindicclemuub.u	. . 3 |- ( ph -> C e. U )
2		eleq1 2259	. . . . 5 |- ( r = C -> ( r e. U <-> C e. U ) )
3		breq2 4037	. . . . . 6 |- ( r = C -> ( q < r <-> q < C ) )
4	3	rexbidv 2498	. . . . 5 |- ( r = C -> ( E. q e. U q < r <-> E. q e. U q < C ) )
5	2, 4	bibi12d 235	. . . 4 |- ( r = C -> ( ( r e. U <-> E. q e. U q < r ) <-> ( C e. U <-> E. q e. U q < C ) ) )
6		dedekindicc.ur	. . . 4 |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )
7		dedekindicc.uss	. . . . 5 |- ( ph -> U C_ ( A [,] B ) )
8	7, 1	sseldd 3184	. . . 4 |- ( ph -> C e. ( A [,] B ) )
9	5, 6, 8	rspcdva 2873	. . 3 |- ( ph -> ( C e. U <-> E. q e. U q < C ) )
10	1, 9	mpbid 147	. 2 |- ( ph -> E. q e. U q < C )
11		dedekindicc.a	. . . . . . 7 |- ( ph -> A e. RR )
12		dedekindicc.b	. . . . . . 7 |- ( ph -> B e. RR )
13		iccssre 10030	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
14	11, 12, 13	syl2anc 411	. . . . . 6 |- ( ph -> ( A [,] B ) C_ RR )
15	14	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> ( A [,] B ) C_ RR )
16		dedekindicc.lss	. . . . . . 7 |- ( ph -> L C_ ( A [,] B ) )
17	16	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> L C_ ( A [,] B ) )
18		simpr 110	. . . . . 6 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z e. L )
19	17, 18	sseldd 3184	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z e. ( A [,] B ) )
20	15, 19	sseldd 3184	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z e. RR )
21	7	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> U C_ ( A [,] B ) )
22		simplrl 535	. . . . . 6 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q e. U )
23	21, 22	sseldd 3184	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q e. ( A [,] B ) )
24	15, 23	sseldd 3184	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q e. RR )
25	14, 8	sseldd 3184	. . . . 5 |- ( ph -> C e. RR )
26	25	ad2antrr 488	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> C e. RR )
27		breq1 4036	. . . . . . . . . 10 |- ( a = q -> ( a < z <-> q < z ) )
28	27	rspcev 2868	. . . . . . . . 9 |- ( ( q e. U /\ q < z ) -> E. a e. U a < z )
29	22, 28	sylan 283	. . . . . . . 8 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> E. a e. U a < z )
30	27	cbvrexv 2730	. . . . . . . 8 |- ( E. a e. U a < z <-> E. q e. U q < z )
31	29, 30	sylib 122	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> E. q e. U q < z )
32		eleq1 2259	. . . . . . . . 9 |- ( r = z -> ( r e. U <-> z e. U ) )
33		breq2 4037	. . . . . . . . . 10 |- ( r = z -> ( q < r <-> q < z ) )
34	33	rexbidv 2498	. . . . . . . . 9 |- ( r = z -> ( E. q e. U q < r <-> E. q e. U q < z ) )
35	32, 34	bibi12d 235	. . . . . . . 8 |- ( r = z -> ( ( r e. U <-> E. q e. U q < r ) <-> ( z e. U <-> E. q e. U q < z ) ) )
36	6	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )
37	19	adantr 276	. . . . . . . 8 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> z e. ( A [,] B ) )
38	35, 36, 37	rspcdva 2873	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> ( z e. U <-> E. q e. U q < z ) )
39	31, 38	mpbird 167	. . . . . 6 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> z e. U )
40		simplll 533	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> ph )
41	18	adantr 276	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> z e. L )
42		dedekindicc.disj	. . . . . . . . 9 |- ( ph -> ( L i^i U ) = (/) )
43		disj 3499	. . . . . . . . 9 |- ( ( L i^i U ) = (/) <-> A. z e. L -. z e. U )
44	42, 43	sylib 122	. . . . . . . 8 |- ( ph -> A. z e. L -. z e. U )
45	44	r19.21bi 2585	. . . . . . 7 |- ( ( ph /\ z e. L ) -> -. z e. U )
46	40, 41, 45	syl2anc 411	. . . . . 6 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> -. z e. U )
47	39, 46	pm2.65da 662	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> -. q < z )
48	20, 24, 47	nltled 8147	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z <_ q )
49		simplrr 536	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q < C )
50	20, 24, 26, 48, 49	lelttrd 8151	. . 3 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z < C )
51	50	ralrimiva 2570	. 2 |- ( ( ph /\ ( q e. U /\ q < C ) ) -> A. z e. L z < C )
52	10, 51	rexlimddv 2619	1 |- ( ph -> A. z e. L z < C )

Syntax hints:   -. wn 3   -> 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:  dedekindicclemub  14863  dedekindicclemloc  14864