Theorem dedekindicclemuub 15650

Description: Lemma for dedekindicc 15657. 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 2301	. . . . 5 |- ( r = C -> ( r e. U <-> C e. U ) )
3		breq2 4129	. . . . . 6 |- ( r = C -> ( q < r <-> q < C ) )
4	3	rexbidv 2551	. . . . 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 3249	. . . 4 |- ( ph -> C e. ( A [,] B ) )
9	5, 6, 8	rspcdva 2934	. . 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 10336	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
14	11, 12, 13	syl2anc 415	. . . . . 6 |- ( ph -> ( A [,] B ) C_ RR )
15	14	ad2antrr 492	. . . . 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 492	. . . . . 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 3249	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z e. ( A [,] B ) )
20	15, 19	sseldd 3249	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z e. RR )
21	7	ad2antrr 492	. . . . . 6 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> U C_ ( A [,] B ) )
22		simplrl 541	. . . . . 6 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q e. U )
23	21, 22	sseldd 3249	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q e. ( A [,] B ) )
24	15, 23	sseldd 3249	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q e. RR )
25	14, 8	sseldd 3249	. . . . 5 |- ( ph -> C e. RR )
26	25	ad2antrr 492	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> C e. RR )
27		breq1 4128	. . . . . . . . . 10 |- ( a = q -> ( a < z <-> q < z ) )
28	27	rspcev 2929	. . . . . . . . 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 2787	. . . . . . . 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 2301	. . . . . . . . 9 |- ( r = z -> ( r e. U <-> z e. U ) )
33		breq2 4129	. . . . . . . . . 10 |- ( r = z -> ( q < r <-> q < z ) )
34	33	rexbidv 2551	. . . . . . . . 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 496	. . . . . . . 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 2934	. . . . . . 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 539	. . . . . . 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 3572	. . . . . . . . 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 2638	. . . . . . 7 |- ( ( ph /\ z e. L ) -> -. z e. U )
46	40, 41, 45	syl2anc 415	. . . . . 6 |- ( ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) /\ q < z ) -> -. z e. U )
47	39, 46	pm2.65da 671	. . . . 5 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> -. q < z )
48	20, 24, 47	nltled 8437	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z <_ q )
49		simplrr 542	. . . 4 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> q < C )
50	20, 24, 26, 48, 49	lelttrd 8441	. . 3 |- ( ( ( ph /\ ( q e. U /\ q < C ) ) /\ z e. L ) -> z < C )
51	50	ralrimiva 2623	. 2 |- ( ( ph /\ ( q e. U /\ q < C ) ) -> A. z e. L z < C )
52	10, 51	rexlimddv 2673	1 |- ( ph -> A. z e. L z < C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   i^i cin 3219   C_ wss 3220   (/)c0 3520   class class class wbr 4125  (class class class)co 6075   RRcr 8168   < clt 8350   [,]cicc 10272

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-icc 10276

This theorem is referenced by:  dedekindicclemub  15651  dedekindicclemloc  15652