Theorem fz01or 10113

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 9576	. . . . . 6 |- 1 e. ( ZZ>= ` 0 )
2		eluzfz1 10033	. . . . . 6 |- ( 1 e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... 1 ) )
3	1, 2	ax-mp 5	. . . . 5 |- 0 e. ( 0 ... 1 )
4		fzsplit 10053	. . . . 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 2244	. . 3 |- ( A e. ( 0 ... 1 ) <-> A e. ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... 1 ) ) )
7		elun 3278	. . 3 |- ( A e. ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... 1 ) ) <-> ( A e. ( 0 ... 0 ) \/ A e. ( ( 0 + 1 ) ... 1 ) ) )
8	6, 7	bitri 184	. 2 |- ( A e. ( 0 ... 1 ) <-> ( A e. ( 0 ... 0 ) \/ A e. ( ( 0 + 1 ) ... 1 ) ) )
9		elfz1eq 10037	. . . 4 |- ( A e. ( 0 ... 0 ) -> A = 0 )
10		0nn0 9193	. . . . . . 7 |- 0 e. NN0
11		nn0uz 9564	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
12	10, 11	eleqtri 2252	. . . . . 6 |- 0 e. ( ZZ>= ` 0 )
13		eluzfz1 10033	. . . . . 6 |- ( 0 e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... 0 ) )
14	12, 13	ax-mp 5	. . . . 5 |- 0 e. ( 0 ... 0 )
15		eleq1 2240	. . . . 5 |- ( A = 0 -> ( A e. ( 0 ... 0 ) <-> 0 e. ( 0 ... 0 ) ) )
16	14, 15	mpbiri 168	. . . 4 |- ( A = 0 -> A e. ( 0 ... 0 ) )
17	9, 16	impbii 126	. . 3 |- ( A e. ( 0 ... 0 ) <-> A = 0 )
18		0p1e1 9035	. . . . . 6 |- ( 0 + 1 ) = 1
19	18	oveq1i 5887	. . . . 5 |- ( ( 0 + 1 ) ... 1 ) = ( 1 ... 1 )
20	19	eleq2i 2244	. . . 4 |- ( A e. ( ( 0 + 1 ) ... 1 ) <-> A e. ( 1 ... 1 ) )
21		elfz1eq 10037	. . . . 5 |- ( A e. ( 1 ... 1 ) -> A = 1 )
22		1nn 8932	. . . . . . . 8 |- 1 e. NN
23		nnuz 9565	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
24	22, 23	eleqtri 2252	. . . . . . 7 |- 1 e. ( ZZ>= ` 1 )
25		eluzfz1 10033	. . . . . . 7 |- ( 1 e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... 1 ) )
26	24, 25	ax-mp 5	. . . . . 6 |- 1 e. ( 1 ... 1 )
27		eleq1 2240	. . . . . 6 |- ( A = 1 -> ( A e. ( 1 ... 1 ) <-> 1 e. ( 1 ... 1 ) ) )
28	26, 27	mpbiri 168	. . . . 5 |- ( A = 1 -> A e. ( 1 ... 1 ) )
29	21, 28	impbii 126	. . . 4 |- ( A e. ( 1 ... 1 ) <-> A = 1 )
30	20, 29	bitri 184	. . 3 |- ( A e. ( ( 0 + 1 ) ... 1 ) <-> A = 1 )
31	17, 30	orbi12i 764	. 2 |- ( ( A e. ( 0 ... 0 ) \/ A e. ( ( 0 + 1 ) ... 1 ) ) <-> ( A = 0 \/ A = 1 ) )
32	8, 31	bitri 184	1 |- ( A e. ( 0 ... 1 ) <-> ( A = 0 \/ A = 1 ) )

Syntax hints:   <-> wb 105   \/ wo 708   = wceq 1353   e. wcel 2148   u. cun 3129   ` cfv 5218  (class class class)co 5877   0cc0 7813   1c1 7814   + caddc 7816   NNcn 8921   NN0cn0 9178   ZZ>=cuz 9530   ...cfz 10010

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-fz 10011

This theorem is referenced by:  hashfiv01gt1  10764  mod2eq1n2dvds  11886