Theorem prm23lt5 13020

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 12866	. . . . 5 |- ( P e. Prime -> P e. NN )
2	1	nnnn0d 9599	. . . 4 |- ( P e. Prime -> P e. NN0 )
3	2	adantr 276	. . 3 |- ( ( P e. Prime /\ P < 5 ) -> P e. NN0 )
4		4nn0 9561	. . . 4 |- 4 e. NN0
5	4	a1i 9	. . 3 |- ( ( P e. Prime /\ P < 5 ) -> 4 e. NN0 )
6		df-5 9345	. . . . . 6 |- 5 = ( 4 + 1 )
7	6	breq2i 4133	. . . . 5 |- ( P < 5 <-> P < ( 4 + 1 ) )
8		prmz 12867	. . . . . . 7 |- ( P e. Prime -> P e. ZZ )
9		4z 9653	. . . . . . 7 |- 4 e. ZZ
10		zleltp1 9679	. . . . . . 7 |- ( ( P e. ZZ /\ 4 e. ZZ ) -> ( P <_ 4 <-> P < ( 4 + 1 ) ) )
11	8, 9, 10	sylancl 417	. . . . . 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 10497	. . 3 |- ( P e. ( 0 ... 4 ) <-> ( P e. NN0 /\ 4 e. NN0 /\ P <_ 4 ) )
16	3, 5, 14, 15	syl3anbrc 1212	. 2 |- ( ( P e. Prime /\ P < 5 ) -> P e. ( 0 ... 4 ) )
17		fz0to4untppr 10509	. . . 4 |- ( 0 ... 4 ) = ( { 0 , 1 , 2 } u. { 3 , 4 } )
18	17	eleq2i 2305	. . 3 |- ( P e. ( 0 ... 4 ) <-> P e. ( { 0 , 1 , 2 } u. { 3 , 4 } ) )
19		elun 3370	. . . . . 6 |- ( P e. ( { 0 , 1 , 2 } u. { 3 , 4 } ) <-> ( P e. { 0 , 1 , 2 } \/ P e. { 3 , 4 } ) )
20		eltpi 3752	. . . . . . . 8 |- ( P e. { 0 , 1 , 2 } -> ( P = 0 \/ P = 1 \/ P = 2 ) )
21		nnne0 9311	. . . . . . . . . . 11 |- ( P e. NN -> P =/= 0 )
22		eqneqall 2430	. . . . . . . . . . . 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 2301	. . . . . . . . . 10 |- ( P = 1 -> ( P e. Prime <-> 1 e. Prime ) )
27		1nprm 12870	. . . . . . . . . . 11 |- -. 1 e. Prime
28	27	pm2.21i 655	. . . . . . . . . 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 724	. . . . . . . . . 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 1344	. . . . . . . 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 3728	. . . . . . . 8 |- ( P e. { 3 , 4 } -> ( P = 3 \/ P = 4 ) )
35		olc 723	. . . . . . . . . 10 |- ( P = 3 -> ( P = 2 \/ P = 3 ) )
36	35	a1d 22	. . . . . . . . 9 |- ( P = 3 -> ( P e. Prime -> ( P = 2 \/ P = 3 ) ) )
37		eleq1 2301	. . . . . . . . . 10 |- ( P = 4 -> ( P e. Prime <-> 4 e. Prime ) )
38		4nprm 12885	. . . . . . . . . . 11 |- -. 4 e. Prime
39	38	pm2.21i 655	. . . . . . . . . 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 728	. . . . . . . 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 728	. . . . . 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 720   \/ w3o 1008   = wceq 1402   e. wcel 2209   =/= wne 2420   u. cun 3218   {cpr 3706   {ctp 3707   class class class wbr 4125  (class class class)co 6075   0cc0 8169   1c1 8170   + caddc 8172   < clt 8350   <_ cle 8351   NNcn 9283   2c2 9334   3c3 9335   4c4 9336   5c5 9337   NN0cn0 9542   ZZcz 9623   ...cfz 10390   Primecprime 12863

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-tp 3713  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-2o 6678  df-er 6797  df-en 7013  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-dvds 12533  df-prm 12864

This theorem is referenced by:  prm23ge5  13021