Theorem dedekindicclemeu 15442

Description: Lemma for dedekindicc 15444. Part of proving uniqueness. (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 ) ) )
dedekindicc.ab	|- ( ph -> A < B )
dedekindicclemeu.are	|- ( ph -> C e. ( A [,] B ) )
dedekindicclemeu.ac	|- ( ph -> ( A. q e. L q < C /\ A. r e. U C < r ) )
dedekindicclemeu.bre	|- ( ph -> D e. ( A [,] B ) )
dedekindicclemeu.bc	|- ( ph -> ( A. q e. L q < D /\ A. r e. U D < r ) )
dedekindicclemeu.lt	|- ( ph -> C < D )

Assertion

Ref	Expression
dedekindicclemeu	|- ( ph -> F. )

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

Allowed substitution hints:   ph( r, q)   D( q)

Proof of Theorem dedekindicclemeu

Step	Hyp	Ref	Expression
1		breq1 4096	. . . 4 |- ( q = C -> ( q < C <-> C < C ) )
2		dedekindicclemeu.ac	. . . . . 6 |- ( ph -> ( A. q e. L q < C /\ A. r e. U C < r ) )
3	2	simpld 112	. . . . 5 |- ( ph -> A. q e. L q < C )
4	3	adantr 276	. . . 4 |- ( ( ph /\ C e. L ) -> A. q e. L q < C )
5		simpr 110	. . . 4 |- ( ( ph /\ C e. L ) -> C e. L )
6	1, 4, 5	rspcdva 2916	. . 3 |- ( ( ph /\ C e. L ) -> C < C )
7		dedekindicc.a	. . . . . . 7 |- ( ph -> A e. RR )
8		dedekindicc.b	. . . . . . 7 |- ( ph -> B e. RR )
9		iccssre 10251	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
10	7, 8, 9	syl2anc 411	. . . . . 6 |- ( ph -> ( A [,] B ) C_ RR )
11		dedekindicclemeu.are	. . . . . 6 |- ( ph -> C e. ( A [,] B ) )
12	10, 11	sseldd 3229	. . . . 5 |- ( ph -> C e. RR )
13	12	ltnrd 8350	. . . 4 |- ( ph -> -. C < C )
14	13	adantr 276	. . 3 |- ( ( ph /\ C e. L ) -> -. C < C )
15	6, 14	pm2.21fal 1418	. 2 |- ( ( ph /\ C e. L ) -> F. )
16		breq2 4097	. . . 4 |- ( r = D -> ( D < r <-> D < D ) )
17		dedekindicclemeu.bc	. . . . . 6 |- ( ph -> ( A. q e. L q < D /\ A. r e. U D < r ) )
18	17	simprd 114	. . . . 5 |- ( ph -> A. r e. U D < r )
19	18	adantr 276	. . . 4 |- ( ( ph /\ D e. U ) -> A. r e. U D < r )
20		simpr 110	. . . 4 |- ( ( ph /\ D e. U ) -> D e. U )
21	16, 19, 20	rspcdva 2916	. . 3 |- ( ( ph /\ D e. U ) -> D < D )
22		dedekindicclemeu.bre	. . . . . 6 |- ( ph -> D e. ( A [,] B ) )
23	10, 22	sseldd 3229	. . . . 5 |- ( ph -> D e. RR )
24	23	ltnrd 8350	. . . 4 |- ( ph -> -. D < D )
25	24	adantr 276	. . 3 |- ( ( ph /\ D e. U ) -> -. D < D )
26	21, 25	pm2.21fal 1418	. 2 |- ( ( ph /\ D e. U ) -> F. )
27		dedekindicclemeu.lt	. . 3 |- ( ph -> C < D )
28		breq2 4097	. . . . 5 |- ( r = D -> ( C < r <-> C < D ) )
29		eleq1 2294	. . . . . 6 |- ( r = D -> ( r e. U <-> D e. U ) )
30	29	orbi2d 798	. . . . 5 |- ( r = D -> ( ( C e. L \/ r e. U ) <-> ( C e. L \/ D e. U ) ) )
31	28, 30	imbi12d 234	. . . 4 |- ( r = D -> ( ( C < r -> ( C e. L \/ r e. U ) ) <-> ( C < D -> ( C e. L \/ D e. U ) ) ) )
32		breq1 4096	. . . . . . 7 |- ( q = C -> ( q < r <-> C < r ) )
33		eleq1 2294	. . . . . . . 8 |- ( q = C -> ( q e. L <-> C e. L ) )
34	33	orbi1d 799	. . . . . . 7 |- ( q = C -> ( ( q e. L \/ r e. U ) <-> ( C e. L \/ r e. U ) ) )
35	32, 34	imbi12d 234	. . . . . 6 |- ( q = C -> ( ( q < r -> ( q e. L \/ r e. U ) ) <-> ( C < r -> ( C e. L \/ r e. U ) ) ) )
36	35	ralbidv 2533	. . . . 5 |- ( q = C -> ( A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) <-> A. r e. ( A [,] B ) ( C < r -> ( C e. L \/ r e. U ) ) ) )
37		dedekindicc.loc	. . . . 5 |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )
38	36, 37, 11	rspcdva 2916	. . . 4 |- ( ph -> A. r e. ( A [,] B ) ( C < r -> ( C e. L \/ r e. U ) ) )
39	31, 38, 22	rspcdva 2916	. . 3 |- ( ph -> ( C < D -> ( C e. L \/ D e. U ) ) )
40	27, 39	mpd 13	. 2 |- ( ph -> ( C e. L \/ D e. U ) )
41	15, 26, 40	mpjaodan 806	1 |- ( ph -> F. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   F. wfal 1403   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 8091   < clt 8273   [,]cicc 10187

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 8183  ax-resscn 8184  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206

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-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 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-icc 10191

This theorem is referenced by:  dedekindicclemicc  15443