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Theorem fz01or 10094
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
StepHypRef Expression
1 1eluzge0 9560 . . . . . 6  |-  1  e.  ( ZZ>= `  0 )
2 eluzfz1 10014 . . . . . 6  |-  ( 1  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... 1
) )
31, 2ax-mp 5 . . . . 5  |-  0  e.  ( 0 ... 1
)
4 fzsplit 10034 . . . . 5  |-  ( 0  e.  ( 0 ... 1 )  ->  (
0 ... 1 )  =  ( ( 0 ... 0 )  u.  (
( 0  +  1 ) ... 1 ) ) )
53, 4ax-mp 5 . . . 4  |-  ( 0 ... 1 )  =  ( ( 0 ... 0 )  u.  (
( 0  +  1 ) ... 1 ) )
65eleq2i 2244 . . 3  |-  ( A  e.  ( 0 ... 1 )  <->  A  e.  ( ( 0 ... 0 )  u.  (
( 0  +  1 ) ... 1 ) ) )
7 elun 3276 . . 3  |-  ( A  e.  ( ( 0 ... 0 )  u.  ( ( 0  +  1 ) ... 1
) )  <->  ( A  e.  ( 0 ... 0
)  \/  A  e.  ( ( 0  +  1 ) ... 1
) ) )
86, 7bitri 184 . 2  |-  ( A  e.  ( 0 ... 1 )  <->  ( A  e.  ( 0 ... 0
)  \/  A  e.  ( ( 0  +  1 ) ... 1
) ) )
9 elfz1eq 10018 . . . 4  |-  ( A  e.  ( 0 ... 0 )  ->  A  =  0 )
10 0nn0 9177 . . . . . . 7  |-  0  e.  NN0
11 nn0uz 9548 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
1210, 11eleqtri 2252 . . . . . 6  |-  0  e.  ( ZZ>= `  0 )
13 eluzfz1 10014 . . . . . 6  |-  ( 0  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... 0
) )
1412, 13ax-mp 5 . . . . 5  |-  0  e.  ( 0 ... 0
)
15 eleq1 2240 . . . . 5  |-  ( A  =  0  ->  ( A  e.  ( 0 ... 0 )  <->  0  e.  ( 0 ... 0
) ) )
1614, 15mpbiri 168 . . . 4  |-  ( A  =  0  ->  A  e.  ( 0 ... 0
) )
179, 16impbii 126 . . 3  |-  ( A  e.  ( 0 ... 0 )  <->  A  = 
0 )
18 0p1e1 9019 . . . . . 6  |-  ( 0  +  1 )  =  1
1918oveq1i 5879 . . . . 5  |-  ( ( 0  +  1 ) ... 1 )  =  ( 1 ... 1
)
2019eleq2i 2244 . . . 4  |-  ( A  e.  ( ( 0  +  1 ) ... 1 )  <->  A  e.  ( 1 ... 1
) )
21 elfz1eq 10018 . . . . 5  |-  ( A  e.  ( 1 ... 1 )  ->  A  =  1 )
22 1nn 8916 . . . . . . . 8  |-  1  e.  NN
23 nnuz 9549 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
2422, 23eleqtri 2252 . . . . . . 7  |-  1  e.  ( ZZ>= `  1 )
25 eluzfz1 10014 . . . . . . 7  |-  ( 1  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... 1
) )
2624, 25ax-mp 5 . . . . . 6  |-  1  e.  ( 1 ... 1
)
27 eleq1 2240 . . . . . 6  |-  ( A  =  1  ->  ( A  e.  ( 1 ... 1 )  <->  1  e.  ( 1 ... 1
) ) )
2826, 27mpbiri 168 . . . . 5  |-  ( A  =  1  ->  A  e.  ( 1 ... 1
) )
2921, 28impbii 126 . . . 4  |-  ( A  e.  ( 1 ... 1 )  <->  A  = 
1 )
3020, 29bitri 184 . . 3  |-  ( A  e.  ( ( 0  +  1 ) ... 1 )  <->  A  = 
1 )
3117, 30orbi12i 764 . 2  |-  ( ( A  e.  ( 0 ... 0 )  \/  A  e.  ( ( 0  +  1 ) ... 1 ) )  <-> 
( A  =  0  \/  A  =  1 ) )
328, 31bitri 184 1  |-  ( A  e.  ( 0 ... 1 )  <->  ( A  =  0  \/  A  =  1 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    \/ wo 708    = wceq 1353    e. wcel 2148    u. cun 3127   ` cfv 5212  (class class class)co 5869   0cc0 7799   1c1 7800    + caddc 7802   NNcn 8905   NN0cn0 9162   ZZ>=cuz 9514   ...cfz 9992
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 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7890  ax-resscn 7891  ax-1cn 7892  ax-1re 7893  ax-icn 7894  ax-addcl 7895  ax-addrcl 7896  ax-mulcl 7897  ax-addcom 7899  ax-addass 7901  ax-distr 7903  ax-i2m1 7904  ax-0lt1 7905  ax-0id 7907  ax-rnegex 7908  ax-cnre 7910  ax-pre-ltirr 7911  ax-pre-ltwlin 7912  ax-pre-lttrn 7913  ax-pre-apti 7914  ax-pre-ltadd 7915
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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7981  df-mnf 7982  df-xr 7983  df-ltxr 7984  df-le 7985  df-sub 8117  df-neg 8118  df-inn 8906  df-n0 9163  df-z 9240  df-uz 9515  df-fz 9993
This theorem is referenced by:  hashfiv01gt1  10743  mod2eq1n2dvds  11864
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