Theorem fz01or 9894

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 9372	. . . . . 6 |- 1 e. ( ZZ>= ` 0 )
2		eluzfz1 9814	. . . . . 6 |- ( 1 e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... 1 ) )
3	1, 2	ax-mp 5	. . . . 5 |- 0 e. ( 0 ... 1 )
4		fzsplit 9834	. . . . 5 |- ( 0 e. ( 0 ... 1 ) -> ( 0 ... 1 ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... 1 ) ) )
5	3, 4	ax-mp 5	. . . 4 |- ( 0 ... 1 ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... 1 ) )
6	5	eleq2i 2206	. . 3 |- ( A e. ( 0 ... 1 ) <-> A e. ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... 1 ) ) )
7		elun 3217	. . 3 |- ( A e. ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... 1 ) ) <-> ( A e. ( 0 ... 0 ) \/ A e. ( ( 0 + 1 ) ... 1 ) ) )
8	6, 7	bitri 183	. 2 |- ( A e. ( 0 ... 1 ) <-> ( A e. ( 0 ... 0 ) \/ A e. ( ( 0 + 1 ) ... 1 ) ) )
9		elfz1eq 9818	. . . 4 |- ( A e. ( 0 ... 0 ) -> A = 0 )
10		0nn0 8995	. . . . . . 7 |- 0 e. NN0
11		nn0uz 9363	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
12	10, 11	eleqtri 2214	. . . . . 6 |- 0 e. ( ZZ>= ` 0 )
13		eluzfz1 9814	. . . . . 6 |- ( 0 e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... 0 ) )
14	12, 13	ax-mp 5	. . . . 5 |- 0 e. ( 0 ... 0 )
15		eleq1 2202	. . . . 5 |- ( A = 0 -> ( A e. ( 0 ... 0 ) <-> 0 e. ( 0 ... 0 ) ) )
16	14, 15	mpbiri 167	. . . 4 |- ( A = 0 -> A e. ( 0 ... 0 ) )
17	9, 16	impbii 125	. . 3 |- ( A e. ( 0 ... 0 ) <-> A = 0 )
18		0p1e1 8837	. . . . . 6 |- ( 0 + 1 ) = 1
19	18	oveq1i 5784	. . . . 5 |- ( ( 0 + 1 ) ... 1 ) = ( 1 ... 1 )
20	19	eleq2i 2206	. . . 4 |- ( A e. ( ( 0 + 1 ) ... 1 ) <-> A e. ( 1 ... 1 ) )
21		elfz1eq 9818	. . . . 5 |- ( A e. ( 1 ... 1 ) -> A = 1 )
22		1nn 8734	. . . . . . . 8 |- 1 e. NN
23		nnuz 9364	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
24	22, 23	eleqtri 2214	. . . . . . 7 |- 1 e. ( ZZ>= ` 1 )
25		eluzfz1 9814	. . . . . . 7 |- ( 1 e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... 1 ) )
26	24, 25	ax-mp 5	. . . . . 6 |- 1 e. ( 1 ... 1 )
27		eleq1 2202	. . . . . 6 |- ( A = 1 -> ( A e. ( 1 ... 1 ) <-> 1 e. ( 1 ... 1 ) ) )
28	26, 27	mpbiri 167	. . . . 5 |- ( A = 1 -> A e. ( 1 ... 1 ) )
29	21, 28	impbii 125	. . . 4 |- ( A e. ( 1 ... 1 ) <-> A = 1 )
30	20, 29	bitri 183	. . 3 |- ( A e. ( ( 0 + 1 ) ... 1 ) <-> A = 1 )
31	17, 30	orbi12i 753	. 2 |- ( ( A e. ( 0 ... 0 ) \/ A e. ( ( 0 + 1 ) ... 1 ) ) <-> ( A = 0 \/ A = 1 ) )
32	8, 31	bitri 183	1 |- ( A e. ( 0 ... 1 ) <-> ( A = 0 \/ A = 1 ) )

Syntax hints:   <-> wb 104   \/ wo 697   = wceq 1331   e. wcel 1480   u. cun 3069   ` cfv 5123  (class class class)co 5774   0cc0 7623   1c1 7624   + caddc 7626   NNcn 8723   NN0cn0 8980   ZZ>=cuz 9329   ...cfz 9793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7714  ax-resscn 7715  ax-1cn 7716  ax-1re 7717  ax-icn 7718  ax-addcl 7719  ax-addrcl 7720  ax-mulcl 7721  ax-addcom 7723  ax-addass 7725  ax-distr 7727  ax-i2m1 7728  ax-0lt1 7729  ax-0id 7731  ax-rnegex 7732  ax-cnre 7734  ax-pre-ltirr 7735  ax-pre-ltwlin 7736  ax-pre-lttrn 7737  ax-pre-apti 7738  ax-pre-ltadd 7739

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7805  df-mnf 7806  df-xr 7807  df-ltxr 7808  df-le 7809  df-sub 7938  df-neg 7939  df-inn 8724  df-n0 8981  df-z 9058  df-uz 9330  df-fz 9794

This theorem is referenced by:  hashfiv01gt1  10531  mod2eq1n2dvds  11579