Theorem fznlem 10237

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 9450	. . . . . . . . . . 11 |- ( M e. ZZ -> M e. RR )
2		zre 9450	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. RR )
3		lenlt 8222	. . . . . . . . . . 11 |- ( ( M e. RR /\ N e. RR ) -> ( M <_ N <-> -. N < M ) )
4	1, 2, 3	syl2an 289	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> -. N < M ) )
5	4	biimpd 144	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N -> -. N < M ) )
6	5	con2d 627	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M -> -. M <_ N ) )
7	6	imp 124	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) -> -. M <_ N )
8	7	adantr 276	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> -. M <_ N )
9		simplll 533	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> M e. ZZ )
10	9	zred 9569	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> M e. RR )
11		simpr 110	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> k e. ZZ )
12	11	zred 9569	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> k e. RR )
13		simpllr 534	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> N e. ZZ )
14	13	zred 9569	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> N e. RR )
15		letr 8229	. . . . . . 7 |- ( ( M e. RR /\ k e. RR /\ N e. RR ) -> ( ( M <_ k /\ k <_ N ) -> M <_ N ) )
16	10, 12, 14, 15	syl3anc 1271	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> ( ( M <_ k /\ k <_ N ) -> M <_ N ) )
17	8, 16	mtod 667	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) /\ k e. ZZ ) -> -. ( M <_ k /\ k <_ N ) )
18	17	ralrimiva 2603	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) -> A. k e. ZZ -. ( M <_ k /\ k <_ N ) )
19		rabeq0 3521	. . . 4 |- ( { k e. ZZ | ( M <_ k /\ k <_ N ) } = (/) <-> A. k e. ZZ -. ( M <_ k /\ k <_ N ) )
20	18, 19	sylibr 134	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) -> { k e. ZZ | ( M <_ k /\ k <_ N ) } = (/) )
21		fzval 10206	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = { k e. ZZ | ( M <_ k /\ k <_ N ) } )
22	21	eqeq1d 2238	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M ... N ) = (/) <-> { k e. ZZ | ( M <_ k /\ k <_ N ) } = (/) ) )
23	22	adantr 276	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) -> ( ( M ... N ) = (/) <-> { k e. ZZ | ( M <_ k /\ k <_ N ) } = (/) ) )
24	20, 23	mpbird 167	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N < M ) -> ( M ... N ) = (/) )
25	24	ex 115	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M -> ( M ... N ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   {crab 2512   (/)c0 3491   class class class wbr 4083  (class class class)co 6001   RRcr 7998   < clt 8181   <_ cle 8182   ZZcz 9446   ...cfz 10204

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-pre-ltwlin 8112

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-neg 8320  df-z 9447  df-fz 10205

This theorem is referenced by:  fzn  10238