Theorem infpn2 13347

Description: There exist infinitely many prime numbers: the set of all primes S is unbounded by infpn 13140, so by unbendc 13345 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 9966	. . . . . . 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 531	. . . . . 6 |- ( ( ( r e. NN /\ 1 < r ) /\ A. m e. NN ( ( r / m ) e. NN -> ( m = 1 \/ m = r ) ) ) -> r e. NN )
4		eluz2b2 10003	. . . . . . . 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 12563	. . . . . . . . 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 2546	. . . . . . 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 477	. . . . . 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 716	. . . . 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 405	. . . . 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 12895	. . . 4 |- ( r e. Prime <-> ( r e. ( ZZ>= ` 2 ) /\ A. m e. NN ( m || r -> ( m = 1 \/ m = r ) ) ) )
14		breq2 4134	. . . . . 6 |- ( n = r -> ( 1 < n <-> 1 < r ) )
15		oveq1 6092	. . . . . . . . 9 |- ( n = r -> ( n / m ) = ( r / m ) )
16	15	eleq1d 2307	. . . . . . . 8 |- ( n = r -> ( ( n / m ) e. NN <-> ( r / m ) e. NN ) )
17		equequ2 1765	. . . . . . . . 9 |- ( n = r -> ( m = n <-> m = r ) )
18	17	orbi2d 802	. . . . . . . 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 2550	. . . . . 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 477	. . . . 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 2985	. . . 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 2235	. 2 |- Prime = S
26		prminf 13346	. 2 |- Prime ~~ NN
27	25, 26	eqbrtrri 4153	1 |- S ~~ NN

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   ~~ cen 7020   1c1 8180   < clt 8360   / cdiv 9002   NNcn 9304   2c2 9355   ZZ>=cuz 9921   || cdvds 12554   Primecprime 12885

This proof depends on 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-13 2211  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-dju 7378  df-inl 7387  df-inr 7388  df-case 7424  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-fac 11164  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-prm 12886

This theorem is used by: (None)