Theorem ivthinclemex 12828

Description: Lemma for ivthinc 12829. 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 3187	. . . 4 |- { w e. ( A [,] B ) | ( F ` w ) < U } C_ ( A [,] B )
5	3, 4	eqsstri 3134	. . 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 3187	. . . 4 |- { w e. ( A [,] B ) | U < ( F ` w ) } C_ ( A [,] B )
9	7, 8	eqsstri 3134	. . 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 12820	. 2 |- ( ph -> E. q e. ( A [,] B ) q e. L )
19	1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7	ivthinclemum 12821	. 2 |- ( ph -> E. r e. ( A [,] B ) r e. R )
20	1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7	ivthinclemlr 12823	. 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 12825	. 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 12826	. 2 |- ( ph -> ( L i^i R ) = (/) )
23	1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7	ivthinclemloc 12827	. 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 12819	1 |- ( ph -> E! z e. ( A (,) B ) ( A. q e. L q < z /\ A. r e. R z < r ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   A.wral 2417   E!wreu 2419   {crab 2421   C_ wss 3076   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   CCcc 7642   RRcr 7643   < clt 7824   (,)cioo 9701   [,]cicc 9704   -cn->ccncf 12765

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764  ax-pre-suploc 7765

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-map 6552  df-sup 6879  df-inf 6880  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-rp 9471  df-ioo 9705  df-icc 9708  df-seqfrec 10250  df-exp 10324  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-cncf 12766

This theorem is referenced by:  ivthinc  12829