Theorem infpn2 13067

Description: There exist infinitely many prime numbers: the set of all primes S is unbounded by infpn 12924, so by unbendc 13065 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 9790	. . . . . . 7 |- ( r e. ( ZZ>= ` 2 ) -> r e. NN )
2	1	adantr 276	. . . . . 6 |- ( ( r e. ( ZZ>= ` 2 ) /\ A. m e. NN ( m || r -> ( m = 1 \/ m = r ) ) ) -> r e. NN )
3		simpll 527	. . . . . 6 |- ( ( ( r e. NN /\ 1 < r ) /\ A. m e. NN ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) -> r e. NN )
4		eluz2b2 9827	. . . . . . . 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 12347	. . . . . . . . 9 |- ( ( r e. NN /\ m e. NN ) -> ( m || r <-> ( r / m ) e. NN ) )
7	6	imbi1d 231	. . . . . . . 8 |- ( ( r e. NN /\ m e. NN ) -> ( ( m || r -> ( m = 1 \/ m = r ) ) <-> ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) )
8	7	ralbidva 2526	. . . . . . 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 473	. . . . . 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 709	. . . . 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 401	. . . . 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 184	. . . 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 12679	. . . 4 |- ( r e. Prime <-> ( r e. ( ZZ>= ` 2 ) /\ A. m e. NN ( m || r -> ( m = 1 \/ m = r ) ) ) )
14		breq2 4090	. . . . . 6 |- ( n = r -> ( 1 < n <-> 1 < r ) )
15		oveq1 6020	. . . . . . . . 9 |- ( n = r -> ( n / m ) = ( r / m ) )
16	15	eleq1d 2298	. . . . . . . 8 |- ( n = r -> ( ( n / m ) e. NN <-> ( r / m ) e. NN ) )
17		equequ2 1759	. . . . . . . . 9 |- ( n = r -> ( m = n <-> m = r ) )
18	17	orbi2d 795	. . . . . . . 8 |- ( n = r -> ( ( m = 1 \/ m = n ) <-> ( m = 1 \/ m = r ) ) )
19	16, 18	imbi12d 234	. . . . . . 7 |- ( n = r -> ( ( ( n / m ) e. NN -> ( m = 1 \/ m = n ) ) <-> ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) )
20	19	ralbidv 2530	. . . . . 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 473	. . . . 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 2963	. . . 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 212	. . 3 |- ( r e. Prime <-> r e. S )
25	24	eqriv 2226	. 2 |- Prime = S
26		prminf 13066	. 2 |- Prime ~~ NN
27	25, 26	eqbrtrri 4109	1 |- S ~~ NN

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   A.wral 2508   {crab 2512   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   ~~ cen 6902   1c1 8023   < clt 8204   / cdiv 8842   NNcn 9133   2c2 9184   ZZ>=cuz 9745   || cdvds 12338   Primecprime 12669

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-2o 6578  df-er 6697  df-pm 6815  df-en 6905  df-dom 6906  df-fin 6907  df-sup 7174  df-inf 7175  df-dju 7228  df-inl 7237  df-inr 7238  df-case 7274  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791  df-fac 10978  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-dvds 12339  df-prm 12670

This theorem is referenced by: (None)