Theorem prm23lt5 12457

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 12303	. . . . 5 |- ( P e. Prime -> P e. NN )
2	1	nnnn0d 9319	. . . 4 |- ( P e. Prime -> P e. NN0 )
3	2	adantr 276	. . 3 |- ( ( P e. Prime /\ P < 5 ) -> P e. NN0 )
4		4nn0 9285	. . . 4 |- 4 e. NN0
5	4	a1i 9	. . 3 |- ( ( P e. Prime /\ P < 5 ) -> 4 e. NN0 )
6		df-5 9069	. . . . . 6 |- 5 = ( 4 + 1 )
7	6	breq2i 4042	. . . . 5 |- ( P < 5 <-> P < ( 4 + 1 ) )
8		prmz 12304	. . . . . . 7 |- ( P e. Prime -> P e. ZZ )
9		4z 9373	. . . . . . 7 |- 4 e. ZZ
10		zleltp1 9398	. . . . . . 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 10204	. . 3 |- ( P e. ( 0 ... 4 ) <-> ( P e. NN0 /\ 4 e. NN0 /\ P <_ 4 ) )
16	3, 5, 14, 15	syl3anbrc 1183	. 2 |- ( ( P e. Prime /\ P < 5 ) -> P e. ( 0 ... 4 ) )
17		fz0to4untppr 10216	. . . 4 |- ( 0 ... 4 ) = ( { 0 , 1 , 2 } u. { 3 , 4 } )
18	17	eleq2i 2263	. . 3 |- ( P e. ( 0 ... 4 ) <-> P e. ( { 0 , 1 , 2 } u. { 3 , 4 } ) )
19		elun 3305	. . . . . 6 |- ( P e. ( { 0 , 1 , 2 } u. { 3 , 4 } ) <-> ( P e. { 0 , 1 , 2 } \/ P e. { 3 , 4 } ) )
20		eltpi 3670	. . . . . . . 8 |- ( P e. { 0 , 1 , 2 } -> ( P = 0 \/ P = 1 \/ P = 2 ) )
21		nnne0 9035	. . . . . . . . . . 11 |- ( P e. NN -> P =/= 0 )
22		eqneqall 2377	. . . . . . . . . . . 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 2259	. . . . . . . . . 10 |- ( P = 1 -> ( P e. Prime <-> 1 e. Prime ) )
27		1nprm 12307	. . . . . . . . . . 11 |- -. 1 e. Prime
28	27	pm2.21i 647	. . . . . . . . . 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 713	. . . . . . . . . 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 1314	. . . . . . . 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 3646	. . . . . . . 8 |- ( P e. { 3 , 4 } -> ( P = 3 \/ P = 4 ) )
35		olc 712	. . . . . . . . . 10 |- ( P = 3 -> ( P = 2 \/ P = 3 ) )
36	35	a1d 22	. . . . . . . . 9 |- ( P = 3 -> ( P e. Prime -> ( P = 2 \/ P = 3 ) ) )
37		eleq1 2259	. . . . . . . . . 10 |- ( P = 4 -> ( P e. Prime <-> 4 e. Prime ) )
38		4nprm 12322	. . . . . . . . . . 11 |- -. 4 e. Prime
39	38	pm2.21i 647	. . . . . . . . . 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 717	. . . . . . . 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 717	. . . . . 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 709   \/ w3o 979   = wceq 1364   e. wcel 2167   =/= wne 2367   u. cun 3155   {cpr 3624   {ctp 3625   class class class wbr 4034  (class class class)co 5925   0cc0 7896   1c1 7897   + caddc 7899   < clt 8078   <_ cle 8079   NNcn 9007   2c2 9058   3c3 9059   4c4 9060   5c5 9061   NN0cn0 9266   ZZcz 9343   ...cfz 10100   Primecprime 12300

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-tp 3631  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-2o 6484  df-er 6601  df-en 6809  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-dvds 11970  df-prm 12301

This theorem is referenced by:  prm23ge5  12458