Theorem fz01or 10278

Description: An integer is in the integer range from zero to one iff it is either zero or one. (Contributed by Jim Kingdon, 11-Nov-2021.)

Assertion

Ref	Expression
fz01or	|- ( A e. ( 0 ... 1 ) <-> ( A = 0 \/ A = 1 ) )

Proof of Theorem fz01or

Step	Hyp	Ref	Expression
1		1eluzge0 8662	. . . . . 6 |- 1 e. ( ZZ>= ` 0 )
2		eluzfz1 9050	. . . . . 6 |- ( 1 e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... 1 ) )
3	1, 2	ax-mp 7	. . . . 5 |- 0 e. ( 0 ... 1 )
4		fzsplit 9070	. . . . 5 |- ( 0 e. ( 0 ... 1 ) -> ( 0 ... 1 ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... 1 ) ) )
5	3, 4	ax-mp 7	. . . 4 |- ( 0 ... 1 ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... 1 ) )
6	5	eleq2i 2145	. . 3 |- ( A e. ( 0 ... 1 ) <-> A e. ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... 1 ) ) )
7		elun 3113	. . 3 |- ( A e. ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... 1 ) ) <-> ( A e. ( 0 ... 0 ) \/ A e. ( ( 0 + 1 ) ... 1 ) ) )
8	6, 7	bitri 182	. 2 |- ( A e. ( 0 ... 1 ) <-> ( A e. ( 0 ... 0 ) \/ A e. ( ( 0 + 1 ) ... 1 ) ) )
9		elfz1eq 9054	. . . 4 |- ( A e. ( 0 ... 0 ) -> A = 0 )
10		0nn0 8303	. . . . . . 7 |- 0 e. NN0
11		nn0uz 8653	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
12	10, 11	eleqtri 2153	. . . . . 6 |- 0 e. ( ZZ>= ` 0 )
13		eluzfz1 9050	. . . . . 6 |- ( 0 e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... 0 ) )
14	12, 13	ax-mp 7	. . . . 5 |- 0 e. ( 0 ... 0 )
15		eleq1 2141	. . . . 5 |- ( A = 0 -> ( A e. ( 0 ... 0 ) <-> 0 e. ( 0 ... 0 ) ) )
16	14, 15	mpbiri 166	. . . 4 |- ( A = 0 -> A e. ( 0 ... 0 ) )
17	9, 16	impbii 124	. . 3 |- ( A e. ( 0 ... 0 ) <-> A = 0 )
18		0p1e1 8153	. . . . . 6 |- ( 0 + 1 ) = 1
19	18	oveq1i 5542	. . . . 5 |- ( ( 0 + 1 ) ... 1 ) = ( 1 ... 1 )
20	19	eleq2i 2145	. . . 4 |- ( A e. ( ( 0 + 1 ) ... 1 ) <-> A e. ( 1 ... 1 ) )
21		elfz1eq 9054	. . . . 5 |- ( A e. ( 1 ... 1 ) -> A = 1 )
22		1nn 8050	. . . . . . . 8 |- 1 e. NN
23		nnuz 8654	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
24	22, 23	eleqtri 2153	. . . . . . 7 |- 1 e. ( ZZ>= ` 1 )
25		eluzfz1 9050	. . . . . . 7 |- ( 1 e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... 1 ) )
26	24, 25	ax-mp 7	. . . . . 6 |- 1 e. ( 1 ... 1 )
27		eleq1 2141	. . . . . 6 |- ( A = 1 -> ( A e. ( 1 ... 1 ) <-> 1 e. ( 1 ... 1 ) ) )
28	26, 27	mpbiri 166	. . . . 5 |- ( A = 1 -> A e. ( 1 ... 1 ) )
29	21, 28	impbii 124	. . . 4 |- ( A e. ( 1 ... 1 ) <-> A = 1 )
30	20, 29	bitri 182	. . 3 |- ( A e. ( ( 0 + 1 ) ... 1 ) <-> A = 1 )
31	17, 30	orbi12i 713	. 2 |- ( ( A e. ( 0 ... 0 ) \/ A e. ( ( 0 + 1 ) ... 1 ) ) <-> ( A = 0 \/ A = 1 ) )
32	8, 31	bitri 182	1 |- ( A e. ( 0 ... 1 ) <-> ( A = 0 \/ A = 1 ) )

Syntax hints:   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   u. cun 2971   ` cfv 4922  (class class class)co 5532   0cc0 6981   1c1 6982   + caddc 6984   NNcn 8039   NN0cn0 8288   ZZ>=cuz 8619   ...cfz 9029

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-fz 9030

This theorem is referenced by:  mod2eq1n2dvds  10279