Theorem fz01or 10496

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 9953	. . . . . 6 |- 1 e. ( ZZ>= ` 0 )
2		eluzfz1 10414	. . . . . 6 |- ( 1 e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... 1 ) )
3	1, 2	ax-mp 5	. . . . 5 |- 0 e. ( 0 ... 1 )
4		fzsplit 10434	. . . . 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 2305	. . 3 |- ( A e. ( 0 ... 1 ) <-> A e. ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... 1 ) ) )
7		elun 3370	. . 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 10418	. . . 4 |- ( A e. ( 0 ... 0 ) -> A = 0 )
10		0nn0 9557	. . . . . . 7 |- 0 e. NN0
11		nn0uz 9936	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
12	10, 11	eleqtri 2313	. . . . . 6 |- 0 e. ( ZZ>= ` 0 )
13		eluzfz1 10414	. . . . . 6 |- ( 0 e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... 0 ) )
14	12, 13	ax-mp 5	. . . . 5 |- 0 e. ( 0 ... 0 )
15		eleq1 2301	. . . . 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 9397	. . . . . 6 |- ( 0 + 1 ) = 1
19	18	oveq1i 6085	. . . . 5 |- ( ( 0 + 1 ) ... 1 ) = ( 1 ... 1 )
20	19	eleq2i 2305	. . . 4 |- ( A e. ( ( 0 + 1 ) ... 1 ) <-> A e. ( 1 ... 1 ) )
21		elfz1eq 10418	. . . . 5 |- ( A e. ( 1 ... 1 ) -> A = 1 )
22		1nn 9294	. . . . . . . 8 |- 1 e. NN
23		nnuz 9937	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
24	22, 23	eleqtri 2313	. . . . . . 7 |- 1 e. ( ZZ>= ` 1 )
25		eluzfz1 10414	. . . . . . 7 |- ( 1 e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... 1 ) )
26	24, 25	ax-mp 5	. . . . . 6 |- 1 e. ( 1 ... 1 )
27		eleq1 2301	. . . . . 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 776	. 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 720   = wceq 1402   e. wcel 2209   u. cun 3218   ` cfv 5372  (class class class)co 6075   0cc0 8169   1c1 8170   + caddc 8172   NNcn 9283   NN0cn0 9542   ZZ>=cuz 9900   ...cfz 10390

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391

This theorem is referenced by:  hashfiv01gt1  11199  mod2eq1n2dvds  12624