Theorem prm23lt5 12619

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 12465	. . . . 5 |- ( P e. Prime -> P e. NN )
2	1	nnnn0d 9350	. . . 4 |- ( P e. Prime -> P e. NN0 )
3	2	adantr 276	. . 3 |- ( ( P e. Prime /\ P < 5 ) -> P e. NN0 )
4		4nn0 9316	. . . 4 |- 4 e. NN0
5	4	a1i 9	. . 3 |- ( ( P e. Prime /\ P < 5 ) -> 4 e. NN0 )
6		df-5 9100	. . . . . 6 |- 5 = ( 4 + 1 )
7	6	breq2i 4053	. . . . 5 |- ( P < 5 <-> P < ( 4 + 1 ) )
8		prmz 12466	. . . . . . 7 |- ( P e. Prime -> P e. ZZ )
9		4z 9404	. . . . . . 7 |- 4 e. ZZ
10		zleltp1 9430	. . . . . . 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 10236	. . 3 |- ( P e. ( 0 ... 4 ) <-> ( P e. NN0 /\ 4 e. NN0 /\ P <_ 4 ) )
16	3, 5, 14, 15	syl3anbrc 1184	. 2 |- ( ( P e. Prime /\ P < 5 ) -> P e. ( 0 ... 4 ) )
17		fz0to4untppr 10248	. . . 4 |- ( 0 ... 4 ) = ( { 0 , 1 , 2 } u. { 3 , 4 } )
18	17	eleq2i 2272	. . 3 |- ( P e. ( 0 ... 4 ) <-> P e. ( { 0 , 1 , 2 } u. { 3 , 4 } ) )
19		elun 3314	. . . . . 6 |- ( P e. ( { 0 , 1 , 2 } u. { 3 , 4 } ) <-> ( P e. { 0 , 1 , 2 } \/ P e. { 3 , 4 } ) )
20		eltpi 3680	. . . . . . . 8 |- ( P e. { 0 , 1 , 2 } -> ( P = 0 \/ P = 1 \/ P = 2 ) )
21		nnne0 9066	. . . . . . . . . . 11 |- ( P e. NN -> P =/= 0 )
22		eqneqall 2386	. . . . . . . . . . . 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 2268	. . . . . . . . . 10 |- ( P = 1 -> ( P e. Prime <-> 1 e. Prime ) )
27		1nprm 12469	. . . . . . . . . . 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 714	. . . . . . . . . 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 1316	. . . . . . . 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 3656	. . . . . . . 8 |- ( P e. { 3 , 4 } -> ( P = 3 \/ P = 4 ) )
35		olc 713	. . . . . . . . . 10 |- ( P = 3 -> ( P = 2 \/ P = 3 ) )
36	35	a1d 22	. . . . . . . . 9 |- ( P = 3 -> ( P e. Prime -> ( P = 2 \/ P = 3 ) ) )
37		eleq1 2268	. . . . . . . . . 10 |- ( P = 4 -> ( P e. Prime <-> 4 e. Prime ) )
38		4nprm 12484	. . . . . . . . . . 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 718	. . . . . . . 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 718	. . . . . 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 710   \/ w3o 980   = wceq 1373   e. wcel 2176   =/= wne 2376   u. cun 3164   {cpr 3634   {ctp 3635   class class class wbr 4045  (class class class)co 5946   0cc0 7927   1c1 7928   + caddc 7930   < clt 8109   <_ cle 8110   NNcn 9038   2c2 9089   3c3 9090   4c4 9091   5c5 9092   NN0cn0 9297   ZZcz 9374   ...cfz 10132   Primecprime 12462

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-tp 3641  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-1o 6504  df-2o 6505  df-er 6622  df-en 6830  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-5 9100  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-fz 10133  df-seqfrec 10595  df-exp 10686  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-dvds 12132  df-prm 12463

This theorem is referenced by:  prm23ge5  12620