Theorem infpn2 12942

Description: There exist infinitely many prime numbers: the set of all primes S is unbounded by infpn 12799, so by unbendc 12940 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 9722	. . . . . . 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 9759	. . . . . . . 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 12222	. . . . . . . . 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 2504	. . . . . . 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 706	. . . . 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 12554	. . . 4 |- ( r e. Prime <-> ( r e. ( ZZ>= ` 2 ) /\ A. m e. NN ( m || r -> ( m = 1 \/ m = r ) ) ) )
14		breq2 4063	. . . . . 6 |- ( n = r -> ( 1 < n <-> 1 < r ) )
15		oveq1 5974	. . . . . . . . 9 |- ( n = r -> ( n / m ) = ( r / m ) )
16	15	eleq1d 2276	. . . . . . . 8 |- ( n = r -> ( ( n / m ) e. NN <-> ( r / m ) e. NN ) )
17		equequ2 1737	. . . . . . . . 9 |- ( n = r -> ( m = n <-> m = r ) )
18	17	orbi2d 792	. . . . . . . 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 2508	. . . . . 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 2939	. . . 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 2204	. 2 |- Prime = S
26		prminf 12941	. 2 |- Prime ~~ NN
27	25, 26	eqbrtrri 4082	1 |- S ~~ NN

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   = wceq 1373   e. wcel 2178   A.wral 2486   {crab 2490   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   ~~ cen 6848   1c1 7961   < clt 8142   / cdiv 8780   NNcn 9071   2c2 9122   ZZ>=cuz 9683   || cdvds 12213   Primecprime 12544

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-2o 6526  df-er 6643  df-pm 6761  df-en 6851  df-dom 6852  df-fin 6853  df-sup 7112  df-inf 7113  df-dju 7166  df-inl 7175  df-inr 7176  df-case 7212  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-fac 10908  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-dvds 12214  df-prm 12545

This theorem is referenced by: (None)