Theorem prm23lt5 12986

Description: A prime less than 5 is either 2 or 3. (Contributed by AV, 5-Jul-2021.)

Assertion

Ref	Expression
prm23lt5	|- ( ( P e. Prime /\ P < 5 ) -> ( P = 2 \/ P = 3 ) )

Proof of Theorem prm23lt5

Step	Hyp	Ref	Expression
1		prmnn 12832	. . . . 5 |- ( P e. Prime -> P e. NN )
2	1	nnnn0d 9570	. . . 4 |- ( P e. Prime -> P e. NN0 )
3	2	adantr 276	. . 3 |- ( ( P e. Prime /\ P < 5 ) -> P e. NN0 )
4		4nn0 9532	. . . 4 |- 4 e. NN0
5	4	a1i 9	. . 3 |- ( ( P e. Prime /\ P < 5 ) -> 4 e. NN0 )
6		df-5 9316	. . . . . 6 |- 5 = ( 4 + 1 )
7	6	breq2i 4122	. . . . 5 |- ( P < 5 <-> P < ( 4 + 1 ) )
8		prmz 12833	. . . . . . 7 |- ( P e. Prime -> P e. ZZ )
9		4z 9624	. . . . . . 7 |- 4 e. ZZ
10		zleltp1 9650	. . . . . . 7 |- ( ( P e. ZZ /\ 4 e. ZZ ) -> ( P <_ 4 <-> P < ( 4 + 1 ) ) )
11	8, 9, 10	sylancl 413	. . . . . 6 |- ( P e. Prime -> ( P <_ 4 <-> P < ( 4 + 1 ) ) )
12	11	biimprd 158	. . . . 5 |- ( P e. Prime -> ( P < ( 4 + 1 ) -> P <_ 4 ) )
13	7, 12	biimtrid 152	. . . 4 |- ( P e. Prime -> ( P < 5 -> P <_ 4 ) )
14	13	imp 124	. . 3 |- ( ( P e. Prime /\ P < 5 ) -> P <_ 4 )
15		elfz2nn0 10468	. . 3 |- ( P e. ( 0 ... 4 ) <-> ( P e. NN0 /\ 4 e. NN0 /\ P <_ 4 ) )
16	3, 5, 14, 15	syl3anbrc 1208	. 2 |- ( ( P e. Prime /\ P < 5 ) -> P e. ( 0 ... 4 ) )
17		fz0to4untppr 10480	. . . 4 |- ( 0 ... 4 ) = ( { 0 , 1 , 2 } u. { 3 , 4 } )
18	17	eleq2i 2301	. . 3 |- ( P e. ( 0 ... 4 ) <-> P e. ( { 0 , 1 , 2 } u. { 3 , 4 } ) )
19		elun 3364	. . . . . 6 |- ( P e. ( { 0 , 1 , 2 } u. { 3 , 4 } ) <-> ( P e. { 0 , 1 , 2 } \/ P e. { 3 , 4 } ) )
20		eltpi 3741	. . . . . . . 8 |- ( P e. { 0 , 1 , 2 } -> ( P = 0 \/ P = 1 \/ P = 2 ) )
21		nnne0 9282	. . . . . . . . . . 11 |- ( P e. NN -> P =/= 0 )
22		eqneqall 2424	. . . . . . . . . . . 12 |- ( P = 0 -> ( P =/= 0 -> ( P = 2 \/ P = 3 ) ) )
23	22	com12 30	. . . . . . . . . . 11 |- ( P =/= 0 -> ( P = 0 -> ( P = 2 \/ P = 3 ) ) )
24	1, 21, 23	3syl 17	. . . . . . . . . 10 |- ( P e. Prime -> ( P = 0 -> ( P = 2 \/ P = 3 ) ) )
25	24	com12 30	. . . . . . . . 9 |- ( P = 0 -> ( P e. Prime -> ( P = 2 \/ P = 3 ) ) )
26		eleq1 2297	. . . . . . . . . 10 |- ( P = 1 -> ( P e. Prime <-> 1 e. Prime ) )
27		1nprm 12836	. . . . . . . . . . 11 |- -. 1 e. Prime
28	27	pm2.21i 651	. . . . . . . . . 10 |- ( 1 e. Prime -> ( P = 2 \/ P = 3 ) )
29	26, 28	biimtrdi 163	. . . . . . . . 9 |- ( P = 1 -> ( P e. Prime -> ( P = 2 \/ P = 3 ) ) )
30		orc 720	. . . . . . . . . 10 |- ( P = 2 -> ( P = 2 \/ P = 3 ) )
31	30	a1d 22	. . . . . . . . 9 |- ( P = 2 -> ( P e. Prime -> ( P = 2 \/ P = 3 ) ) )
32	25, 29, 31	3jaoi 1340	. . . . . . . 8 |- ( ( P = 0 \/ P = 1 \/ P = 2 ) -> ( P e. Prime -> ( P = 2 \/ P = 3 ) ) )
33	20, 32	syl 14	. . . . . . 7 |- ( P e. { 0 , 1 , 2 } -> ( P e. Prime -> ( P = 2 \/ P = 3 ) ) )
34		elpri 3717	. . . . . . . 8 |- ( P e. { 3 , 4 } -> ( P = 3 \/ P = 4 ) )
35		olc 719	. . . . . . . . . 10 |- ( P = 3 -> ( P = 2 \/ P = 3 ) )
36	35	a1d 22	. . . . . . . . 9 |- ( P = 3 -> ( P e. Prime -> ( P = 2 \/ P = 3 ) ) )
37		eleq1 2297	. . . . . . . . . 10 |- ( P = 4 -> ( P e. Prime <-> 4 e. Prime ) )
38		4nprm 12851	. . . . . . . . . . 11 |- -. 4 e. Prime
39	38	pm2.21i 651	. . . . . . . . . 10 |- ( 4 e. Prime -> ( P = 2 \/ P = 3 ) )
40	37, 39	biimtrdi 163	. . . . . . . . 9 |- ( P = 4 -> ( P e. Prime -> ( P = 2 \/ P = 3 ) ) )
41	36, 40	jaoi 724	. . . . . . . 8 |- ( ( P = 3 \/ P = 4 ) -> ( P e. Prime -> ( P = 2 \/ P = 3 ) ) )
42	34, 41	syl 14	. . . . . . 7 |- ( P e. { 3 , 4 } -> ( P e. Prime -> ( P = 2 \/ P = 3 ) ) )
43	33, 42	jaoi 724	. . . . . 6 |- ( ( P e. { 0 , 1 , 2 } \/ P e. { 3 , 4 } ) -> ( P e. Prime -> ( P = 2 \/ P = 3 ) ) )
44	19, 43	sylbi 121	. . . . 5 |- ( P e. ( { 0 , 1 , 2 } u. { 3 , 4 } ) -> ( P e. Prime -> ( P = 2 \/ P = 3 ) ) )
45	44	com12 30	. . . 4 |- ( P e. Prime -> ( P e. ( { 0 , 1 , 2 } u. { 3 , 4 } ) -> ( P = 2 \/ P = 3 ) ) )
46	45	adantr 276	. . 3 |- ( ( P e. Prime /\ P < 5 ) -> ( P e. ( { 0 , 1 , 2 } u. { 3 , 4 } ) -> ( P = 2 \/ P = 3 ) ) )
47	18, 46	biimtrid 152	. 2 |- ( ( P e. Prime /\ P < 5 ) -> ( P e. ( 0 ... 4 ) -> ( P = 2 \/ P = 3 ) ) )
48	16, 47	mpd 13	1 |- ( ( P e. Prime /\ P < 5 ) -> ( P = 2 \/ P = 3 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   \/ w3o 1004   = wceq 1398   e. wcel 2205   =/= wne 2414   u. cun 3212   {cpr 3695   {ctp 3696   class class class wbr 4114  (class class class)co 6058   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   <_ cle 8325   NNcn 9254   2c2 9305   3c3 9306   4c4 9307   5c5 9308   NN0cn0 9513   ZZcz 9594   ...cfz 10361   Primecprime 12829

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-2o 6661  df-er 6780  df-en 6989  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-prm 12830

This theorem is referenced by:  prm23ge5  12987