Theorem ivthinclemex 14503

Description: Lemma for ivthinc 14504. Existence of a number between the lower cut and the upper cut. (Contributed by Jim Kingdon, 20-Feb-2024.)

Hypotheses

Ref	Expression
ivth.1	|- ( ph -> A e. RR )
ivth.2	|- ( ph -> B e. RR )
ivth.3	|- ( ph -> U e. RR )
ivth.4	|- ( ph -> A < B )
ivth.5	|- ( ph -> ( A [,] B ) C_ D )
ivth.7	|- ( ph -> F e. ( D -cn-> CC ) )
ivth.8	|- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
ivth.9	|- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
ivthinc.i	|- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )
ivthinclem.l	|- L = { w e. ( A [,] B ) | ( F ` w ) < U }
ivthinclem.r	|- R = { w e. ( A [,] B ) | U < ( F ` w ) }

Assertion

Ref	Expression
ivthinclemex	|- ( ph -> E! z e. ( A (,) B ) ( A. q e. L q < z /\ A. r e. R z < r ) )

Distinct variable groups:   A, q, r, w   x, A, y, q, r   z, A, q, r   B, q, r, w   x, B, y   z, B   w, F   x, F, y   L, q, r, x, y   z, L   R, q, r, x, y   z, R   w, U   ph, q, r, x, y   ph, z

Allowed substitution hints:   ph( w)   D( x, y, z, w, r, q)   R( w)   U( x, y, z, r, q)   F( z, r, q)   L( w)

Proof of Theorem ivthinclemex

Step	Hyp	Ref	Expression
1		ivth.1	. 2 |- ( ph -> A e. RR )
2		ivth.2	. 2 |- ( ph -> B e. RR )
3		ivthinclem.l	. . . 4 |- L = { w e. ( A [,] B ) | ( F ` w ) < U }
4		ssrab2 3254	. . . 4 |- { w e. ( A [,] B ) | ( F ` w ) < U } C_ ( A [,] B )
5	3, 4	eqsstri 3201	. . 3 |- L C_ ( A [,] B )
6	5	a1i 9	. 2 |- ( ph -> L C_ ( A [,] B ) )
7		ivthinclem.r	. . . 4 |- R = { w e. ( A [,] B ) | U < ( F ` w ) }
8		ssrab2 3254	. . . 4 |- { w e. ( A [,] B ) | U < ( F ` w ) } C_ ( A [,] B )
9	7, 8	eqsstri 3201	. . 3 |- R C_ ( A [,] B )
10	9	a1i 9	. 2 |- ( ph -> R C_ ( A [,] B ) )
11		ivth.3	. . 3 |- ( ph -> U e. RR )
12		ivth.4	. . 3 |- ( ph -> A < B )
13		ivth.5	. . 3 |- ( ph -> ( A [,] B ) C_ D )
14		ivth.7	. . 3 |- ( ph -> F e. ( D -cn-> CC ) )
15		ivth.8	. . 3 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
16		ivth.9	. . 3 |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
17		ivthinc.i	. . 3 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )
18	1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7	ivthinclemlm 14495	. 2 |- ( ph -> E. q e. ( A [,] B ) q e. L )
19	1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7	ivthinclemum 14496	. 2 |- ( ph -> E. r e. ( A [,] B ) r e. R )
20	1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7	ivthinclemlr 14498	. 2 |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )
21	1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7	ivthinclemur 14500	. 2 |- ( ph -> A. r e. ( A [,] B ) ( r e. R <-> E. q e. R q < r ) )
22	1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7	ivthinclemdisj 14501	. 2 |- ( ph -> ( L i^i R ) = (/) )
23	1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7	ivthinclemloc 14502	. 2 |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. R ) ) )
24	1, 2, 6, 10, 18, 19, 20, 21, 22, 23, 12	dedekindicc 14494	1 |- ( ph -> E! z e. ( A (,) B ) ( A. q e. L q < z /\ A. r e. R z < r ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2159   A.wral 2467   E!wreu 2469   {crab 2471   C_ wss 3143   class class class wbr 4017   ` cfv 5230  (class class class)co 5890   CCcc 7826   RRcr 7827   < clt 8009   (,)cioo 9905   [,]cicc 9908   -cn->ccncf 14440

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946  ax-arch 7947  ax-caucvg 7948  ax-pre-suploc 7949

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-po 4310  df-iso 4311  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-isom 5239  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-map 6667  df-sup 7000  df-inf 7001  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-2 8995  df-3 8996  df-4 8997  df-n0 9194  df-z 9271  df-uz 9546  df-rp 9671  df-ioo 9909  df-icc 9912  df-seqfrec 10463  df-exp 10537  df-cj 10868  df-re 10869  df-im 10870  df-rsqrt 11024  df-abs 11025  df-cncf 14441

This theorem is referenced by:  ivthinc  14504