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Theorem nn01to3 11171
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 1167 . . 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 1017 . . . . . . . . 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 998 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  N  <_  3 )
84, 6, 7leltned 9731 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  ( N  <  3  <->  3  =/=  N ) )
9 nesym 2739 . . . . . . . 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 385 . . . . . . . . . . . 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 11151 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( N  e.  NN  /\  N  =/=  1 ) )
16 eluz2 11084 . . . . . . . . . . . . . . . 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 9670 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2  e.  RR  /\  N  e.  RR  /\  2  <_  N )  ->  (
2  <  N  <->  N  =/=  2 ) )
2319, 20, 21, 22syl3an 1270 . . . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . . . . . . . . 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 4459 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  e.  ZZ  ->  ( N  <  3  <->  N  <  ( 2  +  1 ) ) )
3029biimpa 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( N  e.  ZZ  /\  N  <  3 )  ->  N  <  ( 2  +  1 ) )
3130adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( N  e.  ZZ  /\  N  <  3 )  /\  2  <  N
)  ->  N  <  ( 2  +  1 ) )
32 btwnnz 10933 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( 2  e.  ZZ  /\  2  <  N  /\  N  <  ( 2  +  1 ) )  ->  -.  N  e.  ZZ )
3325, 26, 31, 32syl3anc 1228 . . . . . . . . . . . . . . . . . . . . . . . 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 604 . . . . . . . . . . . . . . . . . . . . . 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 1018 . . . . . . . . . . . . . . . . . . 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 2780 . . . . . . . . . . . . . . . 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 429 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  N  <  3 )  ->  N  =  2 )
4443olcd 393 . . . . . . . . . . . . 13  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  N  <  3 )  ->  ( N  =  1  \/  N  =  2 ) )
4544ex 434 . . . . . . . . . . . 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 435 . . . . . . . . . 10  |-  ( N  =/=  1  ->  ( N  e.  NN  ->  ( N  <  3  -> 
( N  =  1  \/  N  =  2 ) ) ) )
4814, 47pm2.61ine 2780 . . . . . . . . 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 1016 . . . . . . 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 430 . . . . 5  |-  ( ( -.  N  =  3  /\  ( N  e. 
NN0  /\  1  <_  N  /\  N  <_  3
) )  ->  ( N  =  1  \/  N  =  2 ) )
5352orcd 392 . . . 4  |-  ( ( -.  N  =  3  /\  ( N  e. 
NN0  /\  1  <_  N  /\  N  <_  3
) )  ->  (
( N  =  1  \/  N  =  2 )  \/  N  =  3 ) )
54 df-3or 974 . . . 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 434 . 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 368    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   RRcr 9487   1c1 9489    + caddc 9491    < clt 9624    <_ cle 9625   NNcn 10532   2c2 10581   3c3 10582   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079
This theorem is referenced by:  hash1to3  12492
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