Theorem infpn2 12613

Description: There exist infinitely many prime numbers: the set of all primes S is unbounded by infpn 12499, so by unbendc 12611 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 9631	. . . . . . 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 9668	. . . . . . . 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 11939	. . . . . . . . 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 2490	. . . . . . 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 705	. . . . 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 12255	. . . 4 |- ( r e. Prime <-> ( r e. ( ZZ>= ` 2 ) /\ A. m e. NN ( m || r -> ( m = 1 \/ m = r ) ) ) )
14		breq2 4033	. . . . . 6 |- ( n = r -> ( 1 < n <-> 1 < r ) )
15		oveq1 5925	. . . . . . . . 9 |- ( n = r -> ( n / m ) = ( r / m ) )
16	15	eleq1d 2262	. . . . . . . 8 |- ( n = r -> ( ( n / m ) e. NN <-> ( r / m ) e. NN ) )
17		equequ2 1724	. . . . . . . . 9 |- ( n = r -> ( m = n <-> m = r ) )
18	17	orbi2d 791	. . . . . . . 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 2494	. . . . . 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 2919	. . . 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 2190	. 2 |- Prime = S
26		prminf 12612	. 2 |- Prime ~~ NN
27	25, 26	eqbrtrri 4052	1 |- S ~~ NN

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2164   A.wral 2472   {crab 2476   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   ~~ cen 6792   1c1 7873   < clt 8054   / cdiv 8691   NNcn 8982   2c2 9033   ZZ>=cuz 9592   || cdvds 11930   Primecprime 12245

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-2o 6470  df-er 6587  df-pm 6705  df-en 6795  df-dom 6796  df-fin 6797  df-sup 7043  df-inf 7044  df-dju 7097  df-inl 7106  df-inr 7107  df-case 7143  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-fac 10797  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-prm 12246

This theorem is referenced by: (None)