Theorem ivthinclemex 14281

Description: Lemma for ivthinc 14282. 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 3242	. . . 4 |- { w e. ( A [,] B ) | ( F ` w ) < U } C_ ( A [,] B )
5	3, 4	eqsstri 3189	. . 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 3242	. . . 4 |- { w e. ( A [,] B ) | U < ( F ` w ) } C_ ( A [,] B )
9	7, 8	eqsstri 3189	. . 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 14273	. 2 |- ( ph -> E. q e. ( A [,] B ) q e. L )
19	1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7	ivthinclemum 14274	. 2 |- ( ph -> E. r e. ( A [,] B ) r e. R )
20	1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7	ivthinclemlr 14276	. 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 14278	. 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 14279	. 2 |- ( ph -> ( L i^i R ) = (/) )
23	1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7	ivthinclemloc 14280	. 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 14272	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 1353   e. wcel 2148   A.wral 2455   E!wreu 2457   {crab 2459   C_ wss 3131   class class class wbr 4005   ` cfv 5218  (class class class)co 5878   CCcc 7812   RRcr 7813   < clt 7995   (,)cioo 9891   [,]cicc 9894   -cn->ccncf 14218

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932  ax-arch 7933  ax-caucvg 7934  ax-pre-suploc 7935

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-map 6653  df-sup 6986  df-inf 6987  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-n0 9180  df-z 9257  df-uz 9532  df-rp 9657  df-ioo 9895  df-icc 9898  df-seqfrec 10449  df-exp 10523  df-cj 10854  df-re 10855  df-im 10856  df-rsqrt 11010  df-abs 11011  df-cncf 14219

This theorem is referenced by:  ivthinc  14282