Theorem fznlem 8905

Description: A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.)

Assertion

Ref	Expression
fznlem	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M -> ( M ... N ) = (/) ) )

Proof of Theorem fznlem

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zre 8249	. . . . . . . . . . 11 |- ( M e. ZZ -> M e. RR )
2		zre 8249	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. RR )
3		lenlt 7094	. . . . . . . . . . 11 |- ( ( M e. RR /\ N e. RR ) -> ( M <_ N <-> -. N < M ) )
4	1, 2, 3	syl2an 273	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> -. N < M ) )
5	4	biimpd 132	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N -> -. N < M ) )
6	5	con2d 554	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M -> -. M <_ N ) )
7	6	imp 115	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) -> -. M <_ N )
8	7	adantr 261	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> -. M <_ N )
9		simplll 485	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> M e. ZZ )
10	9	zred 8360	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> M e. RR )
11		simpr 103	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> k e. ZZ )
12	11	zred 8360	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> k e. RR )
13		simpllr 486	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> N e. ZZ )
14	13	zred 8360	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> N e. RR )
15		letr 7101	. . . . . . 7 |- ( ( M e. RR /\ k e. RR /\ N e. RR ) -> ( ( M <_ k /\ k <_ N ) -> M <_ N ) )
16	10, 12, 14, 15	syl3anc 1135	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> ( ( M <_ k /\ k <_ N ) -> M <_ N ) )
17	8, 16	mtod 589	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> -. ( M <_ k /\ k <_ N ) )
18	17	ralrimiva 2392	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) -> A. k e. ZZ -. ( M <_ k /\ k <_ N ) )
19		rabeq0 3247	. . . 4 |- ( { k e. ZZ | ( M <_ k /\ k <_ N ) } = (/) <-> A. k e. ZZ -. ( M <_ k /\ k <_ N ) )
20	18, 19	sylibr 137	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) -> { k e. ZZ | ( M <_ k /\ k <_ N ) } = (/) )
21		fzval 8876	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = { k e. ZZ | ( M <_ k /\ k <_ N ) } )
22	21	eqeq1d 2048	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M ... N ) = (/) <-> { k e. ZZ | ( M <_ k /\ k <_ N ) } = (/) ) )
23	22	adantr 261	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) -> ( ( M ... N ) = (/) <-> { k e. ZZ | ( M <_ k /\ k <_ N ) } = (/) ) )
24	20, 23	mpbird 156	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) -> ( M ... N ) = (/) )
25	24	ex 108	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M -> ( M ... N ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   A.wral 2306   {crab 2310   (/)c0 3224   class class class wbr 3764  (class class class)co 5512   RRcr 6888   < clt 7060   <_ cle 7061   ZZcz 8245   ...cfz 8874

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltwlin 6997

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-neg 7185  df-z 8246  df-fz 8875

This theorem is referenced by:  fzn  8906