Theorem dedekindicclemeu 12778

Description: Lemma for dedekindicc 12780. 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 3932	. . . 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 111	. . . . 5 |- ( ph -> A. q e. L q < C )
4	3	adantr 274	. . . 4 |- ( ( ph /\ C e. L ) -> A. q e. L q < C )
5		simpr 109	. . . 4 |- ( ( ph /\ C e. L ) -> C e. L )
6	1, 4, 5	rspcdva 2794	. . 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 9738	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
10	7, 8, 9	syl2anc 408	. . . . . 6 |- ( ph -> ( A [,] B ) C_ RR )
11		dedekindicclemeu.are	. . . . . 6 |- ( ph -> C e. ( A [,] B ) )
12	10, 11	sseldd 3098	. . . . 5 |- ( ph -> C e. RR )
13	12	ltnrd 7875	. . . 4 |- ( ph -> -. C < C )
14	13	adantr 274	. . 3 |- ( ( ph /\ C e. L ) -> -. C < C )
15	6, 14	pm2.21fal 1351	. 2 |- ( ( ph /\ C e. L ) -> F. )
16		breq2 3933	. . . 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 113	. . . . 5 |- ( ph -> A. r e. U D < r )
19	18	adantr 274	. . . 4 |- ( ( ph /\ D e. U ) -> A. r e. U D < r )
20		simpr 109	. . . 4 |- ( ( ph /\ D e. U ) -> D e. U )
21	16, 19, 20	rspcdva 2794	. . 3 |- ( ( ph /\ D e. U ) -> D < D )
22		dedekindicclemeu.bre	. . . . . 6 |- ( ph -> D e. ( A [,] B ) )
23	10, 22	sseldd 3098	. . . . 5 |- ( ph -> D e. RR )
24	23	ltnrd 7875	. . . 4 |- ( ph -> -. D < D )
25	24	adantr 274	. . 3 |- ( ( ph /\ D e. U ) -> -. D < D )
26	21, 25	pm2.21fal 1351	. 2 |- ( ( ph /\ D e. U ) -> F. )
27		dedekindicclemeu.lt	. . 3 |- ( ph -> C < D )
28		breq2 3933	. . . . 5 |- ( r = D -> ( C < r <-> C < D ) )
29		eleq1 2202	. . . . . 6 |- ( r = D -> ( r e. U <-> D e. U ) )
30	29	orbi2d 779	. . . . 5 |- ( r = D -> ( ( C e. L \/ r e. U ) <-> ( C e. L \/ D e. U ) ) )
31	28, 30	imbi12d 233	. . . 4 |- ( r = D -> ( ( C < r -> ( C e. L \/ r e. U ) ) <-> ( C < D -> ( C e. L \/ D e. U ) ) ) )
32		breq1 3932	. . . . . . 7 |- ( q = C -> ( q < r <-> C < r ) )
33		eleq1 2202	. . . . . . . 8 |- ( q = C -> ( q e. L <-> C e. L ) )
34	33	orbi1d 780	. . . . . . 7 |- ( q = C -> ( ( q e. L \/ r e. U ) <-> ( C e. L \/ r e. U ) ) )
35	32, 34	imbi12d 233	. . . . . 6 |- ( q = C -> ( ( q < r -> ( q e. L \/ r e. U ) ) <-> ( C < r -> ( C e. L \/ r e. U ) ) ) )
36	35	ralbidv 2437	. . . . 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 2794	. . . 4 |- ( ph -> A. r e. ( A [,] B ) ( C < r -> ( C e. L \/ r e. U ) ) )
39	31, 38, 22	rspcdva 2794	. . 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 787	1 |- ( ph -> F. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   = wceq 1331   F. wfal 1336   e. wcel 1480   A.wral 2416   E.wrex 2417   i^i cin 3070   C_ wss 3071   (/)c0 3363   class class class wbr 3929  (class class class)co 5774   RRcr 7619   < clt 7800   [,]cicc 9674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-icc 9678

This theorem is referenced by:  dedekindicclemicc  12779