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Theorem nn01to3 11264
Description: A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed by Alexander van der Vekens, 13-Sep-2018.)
Assertion
Ref Expression
nn01to3  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  ( N  =  1  \/  N  =  2  \/  N  =  3 ) )

Proof of Theorem nn01to3
StepHypRef Expression
1 3mix3 1180 . . 3  |-  ( N  =  3  ->  ( N  =  1  \/  N  =  2  \/  N  =  3 ) )
21a1d 26 . 2  |-  ( N  =  3  ->  (
( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  -> 
( N  =  1  \/  N  =  2  \/  N  =  3 ) ) )
3 nn0re 10885 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e.  RR )
433ad2ant1 1030 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  N  e.  RR )
5 3re 10690 . . . . . . . . . 10  |-  3  e.  RR
65a1i 11 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  3  e.  RR )
7 simp3 1011 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  N  <_  3 )
84, 6, 7leltned 9793 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  ( N  <  3  <->  3  =/=  N ) )
9 nesym 2682 . . . . . . . 8  |-  ( 3  =/=  N  <->  -.  N  =  3 )
108, 9syl6rbb 266 . . . . . . 7  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  ( -.  N  =  3  <->  N  <  3 ) )
11 elnnnn0c 10922 . . . . . . . . 9  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  1  <_  N ) )
12 orc 387 . . . . . . . . . . . 12  |-  ( N  =  1  ->  ( N  =  1  \/  N  =  2 ) )
1312a1d 26 . . . . . . . . . . 11  |-  ( N  =  1  ->  ( N  <  3  ->  ( N  =  1  \/  N  =  2 ) ) )
1413a1d 26 . . . . . . . . . 10  |-  ( N  =  1  ->  ( N  e.  NN  ->  ( N  <  3  -> 
( N  =  1  \/  N  =  2 ) ) ) )
15 eluz2b3 11239 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( N  e.  NN  /\  N  =/=  1 ) )
16 eluz2 11172 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N ) )
17 2a1 28 . . . . . . . . . . . . . . . . 17  |-  ( N  =  2  ->  (
( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N )  -> 
( N  <  3  ->  N  =  2 ) ) )
18 zre 10948 . . . . . . . . . . . . . . . . . . . 20  |-  ( 2  e.  ZZ  ->  2  e.  RR )
19 zre 10948 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  ZZ  ->  N  e.  RR )
20 id 22 . . . . . . . . . . . . . . . . . . . 20  |-  ( 2  <_  N  ->  2  <_  N )
21 leltne 9728 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2  e.  RR  /\  N  e.  RR  /\  2  <_  N )  ->  (
2  <  N  <->  N  =/=  2 ) )
2218, 19, 20, 21syl3an 1311 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N )  ->  (
2  <  N  <->  N  =/=  2 ) )
23 2z 10976 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  2  e.  ZZ
2423a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( N  e.  ZZ  /\  N  <  3 )  /\  2  <  N
)  ->  2  e.  ZZ )
25 simpr 463 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( N  e.  ZZ  /\  N  <  3 )  /\  2  <  N
)  ->  2  <  N )
26 df-3 10676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  3  =  ( 2  +  1 )
2726a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  ZZ  ->  3  =  ( 2  +  1 ) )
2827breq2d 4417 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  e.  ZZ  ->  ( N  <  3  <->  N  <  ( 2  +  1 ) ) )
2928biimpa 487 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( N  e.  ZZ  /\  N  <  3 )  ->  N  <  ( 2  +  1 ) )
3029adantr 467 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( N  e.  ZZ  /\  N  <  3 )  /\  2  <  N
)  ->  N  <  ( 2  +  1 ) )
31 btwnnz 11019 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( 2  e.  ZZ  /\  2  <  N  /\  N  <  ( 2  +  1 ) )  ->  -.  N  e.  ZZ )
3224, 25, 30, 31syl3anc 1269 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( N  e.  ZZ  /\  N  <  3 )  /\  2  <  N
)  ->  -.  N  e.  ZZ )
3332pm2.21d 110 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  ZZ  /\  N  <  3 )  /\  2  <  N
)  ->  ( N  e.  ZZ  ->  N  = 
2 ) )
3433exp31 609 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  ZZ  ->  ( N  <  3  ->  (
2  <  N  ->  ( N  e.  ZZ  ->  N  =  2 ) ) ) )
3534com24 90 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  ZZ  ->  ( N  e.  ZZ  ->  ( 2  <  N  -> 
( N  <  3  ->  N  =  2 ) ) ) )
3635pm2.43i 49 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  ZZ  ->  (
2  <  N  ->  ( N  <  3  ->  N  =  2 ) ) )
37363ad2ant2 1031 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N )  ->  (
2  <  N  ->  ( N  <  3  ->  N  =  2 ) ) )
3822, 37sylbird 239 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N )  ->  ( N  =/=  2  ->  ( N  <  3  ->  N  =  2 ) ) )
3938com12 32 . . . . . . . . . . . . . . . . 17  |-  ( N  =/=  2  ->  (
( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N )  -> 
( N  <  3  ->  N  =  2 ) ) )
4017, 39pm2.61ine 2709 . . . . . . . . . . . . . . . 16  |-  ( ( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N )  ->  ( N  <  3  ->  N  =  2 ) )
4116, 40sylbi 199 . . . . . . . . . . . . . . 15  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  <  3  ->  N  = 
2 ) )
4241imp 431 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  N  <  3 )  ->  N  =  2 )
4342olcd 395 . . . . . . . . . . . . 13  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  N  <  3 )  ->  ( N  =  1  \/  N  =  2 ) )
4443ex 436 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  <  3  ->  ( N  =  1  \/  N  =  2 ) ) )
4515, 44sylbir 217 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  N  =/=  1 )  -> 
( N  <  3  ->  ( N  =  1  \/  N  =  2 ) ) )
4645expcom 437 . . . . . . . . . 10  |-  ( N  =/=  1  ->  ( N  e.  NN  ->  ( N  <  3  -> 
( N  =  1  \/  N  =  2 ) ) ) )
4714, 46pm2.61ine 2709 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( N  <  3  ->  ( N  =  1  \/  N  =  2 ) ) )
4811, 47sylbir 217 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  1  <_  N )  -> 
( N  <  3  ->  ( N  =  1  \/  N  =  2 ) ) )
49483adant3 1029 . . . . . . 7  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  ( N  <  3  ->  ( N  =  1  \/  N  =  2 ) ) )
5010, 49sylbid 219 . . . . . 6  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  ( -.  N  =  3  ->  ( N  =  1  \/  N  =  2 ) ) )
5150impcom 432 . . . . 5  |-  ( ( -.  N  =  3  /\  ( N  e. 
NN0  /\  1  <_  N  /\  N  <_  3
) )  ->  ( N  =  1  \/  N  =  2 ) )
5251orcd 394 . . . 4  |-  ( ( -.  N  =  3  /\  ( N  e. 
NN0  /\  1  <_  N  /\  N  <_  3
) )  ->  (
( N  =  1  \/  N  =  2 )  \/  N  =  3 ) )
53 df-3or 987 . . . 4  |-  ( ( N  =  1  \/  N  =  2  \/  N  =  3 )  <-> 
( ( N  =  1  \/  N  =  2 )  \/  N  =  3 ) )
5452, 53sylibr 216 . . 3  |-  ( ( -.  N  =  3  /\  ( N  e. 
NN0  /\  1  <_  N  /\  N  <_  3
) )  ->  ( N  =  1  \/  N  =  2  \/  N  =  3 ) )
5554ex 436 . 2  |-  ( -.  N  =  3  -> 
( ( N  e. 
NN0  /\  1  <_  N  /\  N  <_  3
)  ->  ( N  =  1  \/  N  =  2  \/  N  =  3 ) ) )
562, 55pm2.61i 168 1  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  ( N  =  1  \/  N  =  2  \/  N  =  3 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    \/ w3o 985    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   class class class wbr 4405   ` cfv 5585  (class class class)co 6295   RRcr 9543   1c1 9545    + caddc 9547    < clt 9680    <_ cle 9681   NNcn 10616   2c2 10666   3c3 10667   NN0cn0 10876   ZZcz 10944   ZZ>=cuz 11166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167
This theorem is referenced by:  hash1to3  12655
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