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Theorem nn01to3 11176
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 1165 . . 3  |-  ( N  =  3  ->  ( N  =  1  \/  N  =  2  \/  N  =  3 ) )
21a1d 25 . 2  |-  ( N  =  3  ->  (
( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  -> 
( N  =  1  \/  N  =  2  \/  N  =  3 ) ) )
3 nn0re 10800 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e.  RR )
433ad2ant1 1015 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  N  e.  RR )
5 3re 10605 . . . . . . . . . 10  |-  3  e.  RR
65a1i 11 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  3  e.  RR )
7 simp3 996 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  N  <_  3 )
84, 6, 7leltned 9725 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  ( N  <  3  <->  3  =/=  N ) )
9 nesym 2726 . . . . . . . 8  |-  ( 3  =/=  N  <->  -.  N  =  3 )
108, 9syl6rbb 262 . . . . . . 7  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  ( -.  N  =  3  <->  N  <  3 ) )
11 elnnnn0c 10837 . . . . . . . . 9  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  1  <_  N ) )
12 orc 383 . . . . . . . . . . . 12  |-  ( N  =  1  ->  ( N  =  1  \/  N  =  2 ) )
1312a1d 25 . . . . . . . . . . 11  |-  ( N  =  1  ->  ( N  <  3  ->  ( N  =  1  \/  N  =  2 ) ) )
1413a1d 25 . . . . . . . . . 10  |-  ( N  =  1  ->  ( N  e.  NN  ->  ( N  <  3  -> 
( N  =  1  \/  N  =  2 ) ) ) )
15 eluz2b3 11156 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( N  e.  NN  /\  N  =/=  1 ) )
16 eluz2 11088 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N ) )
17 ax-1 6 . . . . . . . . . . . . . . . . . 18  |-  ( N  =  2  ->  ( N  <  3  ->  N  =  2 ) )
1817a1d 25 . . . . . . . . . . . . . . . . 17  |-  ( N  =  2  ->  (
( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N )  -> 
( N  <  3  ->  N  =  2 ) ) )
19 zre 10864 . . . . . . . . . . . . . . . . . . . 20  |-  ( 2  e.  ZZ  ->  2  e.  RR )
20 zre 10864 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  ZZ  ->  N  e.  RR )
21 id 22 . . . . . . . . . . . . . . . . . . . 20  |-  ( 2  <_  N  ->  2  <_  N )
22 leltne 9663 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2  e.  RR  /\  N  e.  RR  /\  2  <_  N )  ->  (
2  <  N  <->  N  =/=  2 ) )
2319, 20, 21, 22syl3an 1268 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N )  ->  (
2  <  N  <->  N  =/=  2 ) )
24 2z 10892 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  2  e.  ZZ
2524a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( N  e.  ZZ  /\  N  <  3 )  /\  2  <  N
)  ->  2  e.  ZZ )
26 simpr 459 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( N  e.  ZZ  /\  N  <  3 )  /\  2  <  N
)  ->  2  <  N )
27 df-3 10591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  3  =  ( 2  +  1 )
2827a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  ZZ  ->  3  =  ( 2  +  1 ) )
2928breq2d 4451 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  e.  ZZ  ->  ( N  <  3  <->  N  <  ( 2  +  1 ) ) )
3029biimpa 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( N  e.  ZZ  /\  N  <  3 )  ->  N  <  ( 2  +  1 ) )
3130adantr 463 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( N  e.  ZZ  /\  N  <  3 )  /\  2  <  N
)  ->  N  <  ( 2  +  1 ) )
32 btwnnz 10935 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( 2  e.  ZZ  /\  2  <  N  /\  N  <  ( 2  +  1 ) )  ->  -.  N  e.  ZZ )
3325, 26, 31, 32syl3anc 1226 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( N  e.  ZZ  /\  N  <  3 )  /\  2  <  N
)  ->  -.  N  e.  ZZ )
3433pm2.21d 106 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  ZZ  /\  N  <  3 )  /\  2  <  N
)  ->  ( N  e.  ZZ  ->  N  = 
2 ) )
3534exp31 602 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  ZZ  ->  ( N  <  3  ->  (
2  <  N  ->  ( N  e.  ZZ  ->  N  =  2 ) ) ) )
3635com24 87 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  ZZ  ->  ( N  e.  ZZ  ->  ( 2  <  N  -> 
( N  <  3  ->  N  =  2 ) ) ) )
3736pm2.43i 47 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  ZZ  ->  (
2  <  N  ->  ( N  <  3  ->  N  =  2 ) ) )
38373ad2ant2 1016 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N )  ->  (
2  <  N  ->  ( N  <  3  ->  N  =  2 ) ) )
3923, 38sylbird 235 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N )  ->  ( N  =/=  2  ->  ( N  <  3  ->  N  =  2 ) ) )
4039com12 31 . . . . . . . . . . . . . . . . 17  |-  ( N  =/=  2  ->  (
( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N )  -> 
( N  <  3  ->  N  =  2 ) ) )
4118, 40pm2.61ine 2767 . . . . . . . . . . . . . . . 16  |-  ( ( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N )  ->  ( N  <  3  ->  N  =  2 ) )
4216, 41sylbi 195 . . . . . . . . . . . . . . 15  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  <  3  ->  N  = 
2 ) )
4342imp 427 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  N  <  3 )  ->  N  =  2 )
4443olcd 391 . . . . . . . . . . . . 13  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  N  <  3 )  ->  ( N  =  1  \/  N  =  2 ) )
4544ex 432 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  <  3  ->  ( N  =  1  \/  N  =  2 ) ) )
4615, 45sylbir 213 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  N  =/=  1 )  -> 
( N  <  3  ->  ( N  =  1  \/  N  =  2 ) ) )
4746expcom 433 . . . . . . . . . 10  |-  ( N  =/=  1  ->  ( N  e.  NN  ->  ( N  <  3  -> 
( N  =  1  \/  N  =  2 ) ) ) )
4814, 47pm2.61ine 2767 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( N  <  3  ->  ( N  =  1  \/  N  =  2 ) ) )
4911, 48sylbir 213 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  1  <_  N )  -> 
( N  <  3  ->  ( N  =  1  \/  N  =  2 ) ) )
50493adant3 1014 . . . . . . 7  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  ( N  <  3  ->  ( N  =  1  \/  N  =  2 ) ) )
5110, 50sylbid 215 . . . . . 6  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  ( -.  N  =  3  ->  ( N  =  1  \/  N  =  2 ) ) )
5251impcom 428 . . . . 5  |-  ( ( -.  N  =  3  /\  ( N  e. 
NN0  /\  1  <_  N  /\  N  <_  3
) )  ->  ( N  =  1  \/  N  =  2 ) )
5352orcd 390 . . . 4  |-  ( ( -.  N  =  3  /\  ( N  e. 
NN0  /\  1  <_  N  /\  N  <_  3
) )  ->  (
( N  =  1  \/  N  =  2 )  \/  N  =  3 ) )
54 df-3or 972 . . . 4  |-  ( ( N  =  1  \/  N  =  2  \/  N  =  3 )  <-> 
( ( N  =  1  \/  N  =  2 )  \/  N  =  3 ) )
5553, 54sylibr 212 . . 3  |-  ( ( -.  N  =  3  /\  ( N  e. 
NN0  /\  1  <_  N  /\  N  <_  3
) )  ->  ( N  =  1  \/  N  =  2  \/  N  =  3 ) )
5655ex 432 . 2  |-  ( -.  N  =  3  -> 
( ( N  e. 
NN0  /\  1  <_  N  /\  N  <_  3
)  ->  ( N  =  1  \/  N  =  2  \/  N  =  3 ) ) )
572, 56pm2.61i 164 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 184    \/ wo 366    /\ wa 367    \/ w3o 970    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   RRcr 9480   1c1 9482    + caddc 9484    < clt 9617    <_ cle 9618   NNcn 10531   2c2 10581   3c3 10582   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083
This theorem is referenced by:  hash1to3  12514
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