Theorem ivthinclemex 15743

Description: Lemma for ivthinc 15744. 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 3333	. . . 4 |- { w e. ( A [,] B ) | ( F ` w ) < U } C_ ( A [,] B )
5	3, 4	eqsstri 3280	. . 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 3333	. . . 4 |- { w e. ( A [,] B ) | U < ( F ` w ) } C_ ( A [,] B )
9	7, 8	eqsstri 3280	. . 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 15735	. 2 |- ( ph -> E. q e. ( A [,] B ) q e. L )
19	1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7	ivthinclemum 15736	. 2 |- ( ph -> E. r e. ( A [,] B ) r e. R )
20	1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7	ivthinclemlr 15738	. 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 15740	. 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 15741	. 2 |- ( ph -> ( L i^i R ) = (/) )
23	1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7	ivthinclemloc 15742	. 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 15734	1 |- ( ph -> E! z e. ( A (,) B ) ( A. q e. L q < z /\ A. r e. R z < r ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   E!wreu 2530   {crab 2532   C_ wss 3220   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   < clt 8360   (,)cioo 10290   [,]cicc 10293   -cn->ccncf 15671

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-pre-suploc 8300

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-map 6924  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-rp 10055  df-ioo 10294  df-icc 10297  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-cncf 15672

This theorem is used by:  ivthinc  15744