Theorem apbtwnz 10288

Description: There is a unique greatest integer less than or equal to a real number which is apart from all integers. (Contributed by Jim Kingdon, 11-May-2022.)

Assertion

Ref	Expression
apbtwnz	|- ( ( A e. RR /\ A. n e. ZZ A # n ) -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )

Distinct variable group:   A, n, x

Proof of Theorem apbtwnz

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . 3 |- ( ( A e. RR /\ A. n e. ZZ A # n ) -> A e. RR )
2		simpr 110	. . . . 5 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ A < m ) -> A < m )
3	2	olcd 735	. . . 4 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ A < m ) -> ( m <_ A \/ A < m ) )
4		simpr 110	. . . . . . . 8 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> m e. ZZ )
5	4	zred 9389	. . . . . . 7 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> m e. RR )
6	5	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ m < A ) -> m e. RR )
7	1	adantr 276	. . . . . . 7 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> A e. RR )
8	7	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ m < A ) -> A e. RR )
9		simpr 110	. . . . . 6 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ m < A ) -> m < A )
10	6, 8, 9	ltled 8090	. . . . 5 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ m < A ) -> m <_ A )
11	10	orcd 734	. . . 4 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ m < A ) -> ( m <_ A \/ A < m ) )
12		breq2 4019	. . . . . 6 |- ( n = m -> ( A # n <-> A # m ) )
13		simplr 528	. . . . . 6 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> A. n e. ZZ A # n )
14	12, 13, 4	rspcdva 2858	. . . . 5 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> A # m )
15		reaplt 8559	. . . . . 6 |- ( ( A e. RR /\ m e. RR ) -> ( A # m <-> ( A < m \/ m < A ) ) )
16	7, 5, 15	syl2anc 411	. . . . 5 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> ( A # m <-> ( A < m \/ m < A ) ) )
17	14, 16	mpbid 147	. . . 4 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> ( A < m \/ m < A ) )
18	3, 11, 17	mpjaodan 799	. . 3 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> ( m <_ A \/ A < m ) )
19	1, 18	exbtwnzlemex 10264	. 2 |- ( ( A e. RR /\ A. n e. ZZ A # n ) -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
20	19, 1	exbtwnz 10265	1 |- ( ( A e. RR /\ A. n e. ZZ A # n ) -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   e. wcel 2158   A.wral 2465   E!wreu 2467   class class class wbr 4015  (class class class)co 5888   RRcr 7824   1c1 7826   + caddc 7828   < clt 8006   <_ cle 8007   # cap 8552   ZZcz 9267

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 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-arch 7944

This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-inn 8934  df-n0 9191  df-z 9268

This theorem is referenced by:  flapcl  10289