Theorem fz01or 10345

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 9807	. . . . . 6 |- 1 e. ( ZZ>= ` 0 )
2		eluzfz1 10265	. . . . . 6 |- ( 1 e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... 1 ) )
3	1, 2	ax-mp 5	. . . . 5 |- 0 e. ( 0 ... 1 )
4		fzsplit 10285	. . . . 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 2298	. . 3 |- ( A e. ( 0 ... 1 ) <-> A e. ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... 1 ) ) )
7		elun 3348	. . 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 10269	. . . 4 |- ( A e. ( 0 ... 0 ) -> A = 0 )
10		0nn0 9416	. . . . . . 7 |- 0 e. NN0
11		nn0uz 9790	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
12	10, 11	eleqtri 2306	. . . . . 6 |- 0 e. ( ZZ>= ` 0 )
13		eluzfz1 10265	. . . . . 6 |- ( 0 e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... 0 ) )
14	12, 13	ax-mp 5	. . . . 5 |- 0 e. ( 0 ... 0 )
15		eleq1 2294	. . . . 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 9256	. . . . . 6 |- ( 0 + 1 ) = 1
19	18	oveq1i 6027	. . . . 5 |- ( ( 0 + 1 ) ... 1 ) = ( 1 ... 1 )
20	19	eleq2i 2298	. . . 4 |- ( A e. ( ( 0 + 1 ) ... 1 ) <-> A e. ( 1 ... 1 ) )
21		elfz1eq 10269	. . . . 5 |- ( A e. ( 1 ... 1 ) -> A = 1 )
22		1nn 9153	. . . . . . . 8 |- 1 e. NN
23		nnuz 9791	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
24	22, 23	eleqtri 2306	. . . . . . 7 |- 1 e. ( ZZ>= ` 1 )
25		eluzfz1 10265	. . . . . . 7 |- ( 1 e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... 1 ) )
26	24, 25	ax-mp 5	. . . . . 6 |- 1 e. ( 1 ... 1 )
27		eleq1 2294	. . . . . 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 771	. 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 715   = wceq 1397   e. wcel 2202   u. cun 3198   ` cfv 5326  (class class class)co 6017   0cc0 8031   1c1 8032   + caddc 8034   NNcn 9142   NN0cn0 9401   ZZ>=cuz 9754   ...cfz 10242

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243

This theorem is referenced by:  hashfiv01gt1  11043  mod2eq1n2dvds  12439