Theorem infpn2 12385

Description: There exist infinitely many prime numbers: the set of all primes S is unbounded by infpn 12287, so by unbendc 12383 it is infinite. This is Metamath 100 proof #11. (Contributed by NM, 5-May-2005.)

Hypothesis

Ref	Expression
infpn2.1	|- S = { n e. NN | ( 1 < n /\ A. m e. NN ( ( n / m ) e. NN -> ( m = 1 \/ m = n ) ) ) }

Assertion

Ref	Expression
infpn2	|- S ~~ NN

Distinct variable group:   m, n

Allowed substitution hints:   S( m, n)

Proof of Theorem infpn2

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluz2nn 9500	. . . . . . 7 |- ( r e. ( ZZ>= ` 2 ) -> r e. NN )
2	1	adantr 274	. . . . . 6 |- ( ( r e. ( ZZ>= ` 2 ) /\ A. m e. NN ( m || r -> ( m = 1 \/ m = r ) ) ) -> r e. NN )
3		simpll 519	. . . . . 6 |- ( ( ( r e. NN /\ 1 < r ) /\ A. m e. NN ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) -> r e. NN )
4		eluz2b2 9537	. . . . . . . 8 |- ( r e. ( ZZ>= ` 2 ) <-> ( r e. NN /\ 1 < r ) )
5	4	a1i 9	. . . . . . 7 |- ( r e. NN -> ( r e. ( ZZ>= ` 2 ) <-> ( r e. NN /\ 1 < r ) ) )
6		nndivdvds 11732	. . . . . . . . 9 |- ( ( r e. NN /\ m e. NN ) -> ( m || r <-> ( r / m ) e. NN ) )
7	6	imbi1d 230	. . . . . . . 8 |- ( ( r e. NN /\ m e. NN ) -> ( ( m || r -> ( m = 1 \/ m = r ) ) <-> ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) )
8	7	ralbidva 2461	. . . . . . 7 |- ( r e. NN -> ( A. m e. NN ( m || r -> ( m = 1 \/ m = r ) ) <-> A. m e. NN ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) )
9	5, 8	anbi12d 465	. . . . . 6 |- ( r e. NN -> ( ( r e. ( ZZ>= ` 2 ) /\ A. m e. NN ( m || r -> ( m = 1 \/ m = r ) ) ) <-> ( ( r e. NN /\ 1 < r ) /\ A. m e. NN ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) ) )
10	2, 3, 9	pm5.21nii 694	. . . . 5 |- ( ( r e. ( ZZ>= ` 2 ) /\ A. m e. NN ( m || r -> ( m = 1 \/ m = r ) ) ) <-> ( ( r e. NN /\ 1 < r ) /\ A. m e. NN ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) )
11		anass 399	. . . . 5 |- ( ( ( r e. NN /\ 1 < r ) /\ A. m e. NN ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) <-> ( r e. NN /\ ( 1 < r /\ A. m e. NN ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) ) )
12	10, 11	bitri 183	. . . 4 |- ( ( r e. ( ZZ>= ` 2 ) /\ A. m e. NN ( m || r -> ( m = 1 \/ m = r ) ) ) <-> ( r e. NN /\ ( 1 < r /\ A. m e. NN ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) ) )
13		isprm2 12045	. . . 4 |- ( r e. Prime <-> ( r e. ( ZZ>= ` 2 ) /\ A. m e. NN ( m || r -> ( m = 1 \/ m = r ) ) ) )
14		breq2 3985	. . . . . 6 |- ( n = r -> ( 1 < n <-> 1 < r ) )
15		oveq1 5848	. . . . . . . . 9 |- ( n = r -> ( n / m ) = ( r / m ) )
16	15	eleq1d 2234	. . . . . . . 8 |- ( n = r -> ( ( n / m ) e. NN <-> ( r / m ) e. NN ) )
17		equequ2 1701	. . . . . . . . 9 |- ( n = r -> ( m = n <-> m = r ) )
18	17	orbi2d 780	. . . . . . . 8 |- ( n = r -> ( ( m = 1 \/ m = n ) <-> ( m = 1 \/ m = r ) ) )
19	16, 18	imbi12d 233	. . . . . . 7 |- ( n = r -> ( ( ( n / m ) e. NN -> ( m = 1 \/ m = n ) ) <-> ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) )
20	19	ralbidv 2465	. . . . . 6 |- ( n = r -> ( A. m e. NN ( ( n / m ) e. NN -> ( m = 1 \/ m = n ) ) <-> A. m e. NN ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) )
21	14, 20	anbi12d 465	. . . . 5 |- ( n = r -> ( ( 1 < n /\ A. m e. NN ( ( n / m ) e. NN -> ( m = 1 \/ m = n ) ) ) <-> ( 1 < r /\ A. m e. NN ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) ) )
22		infpn2.1	. . . . 5 |- S = { n e. NN | ( 1 < n /\ A. m e. NN ( ( n / m ) e. NN -> ( m = 1 \/ m = n ) ) ) }
23	21, 22	elrab2 2884	. . . 4 |- ( r e. S <-> ( r e. NN /\ ( 1 < r /\ A. m e. NN ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) ) )
24	12, 13, 23	3bitr4i 211	. . 3 |- ( r e. Prime <-> r e. S )
25	24	eqriv 2162	. 2 |- Prime = S
26		prminf 12384	. 2 |- Prime ~~ NN
27	25, 26	eqbrtrri 4004	1 |- S ~~ NN

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1343   e. wcel 2136   A.wral 2443   {crab 2447   class class class wbr 3981   ` cfv 5187  (class class class)co 5841   ~~ cen 6700   1c1 7750   < clt 7929   / cdiv 8564   NNcn 8853   2c2 8904   ZZ>=cuz 9462   || cdvds 11723   Primecprime 12035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868  ax-caucvg 7869

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-isom 5196  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-frec 6355  df-1o 6380  df-2o 6381  df-er 6497  df-pm 6613  df-en 6703  df-dom 6704  df-fin 6705  df-sup 6945  df-inf 6946  df-dju 6999  df-inl 7008  df-inr 7009  df-case 7045  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-3 8913  df-4 8914  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-fz 9941  df-fzo 10074  df-fl 10201  df-mod 10254  df-seqfrec 10377  df-exp 10451  df-fac 10635  df-cj 10780  df-re 10781  df-im 10782  df-rsqrt 10936  df-abs 10937  df-dvds 11724  df-prm 12036

This theorem is referenced by: (None)