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| Mirrors > Home > MPE Home > Th. List > 2prm | Structured version Visualization version GIF version | ||
| Description: 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.) |
| Ref | Expression |
|---|---|
| 2prm | ⊢ 2 ∈ ℙ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 2z 12649 | . . 3 ⊢ 2 ∈ ℤ | |
| 2 | 1lt2 12437 | . . 3 ⊢ 1 < 2 | |
| 3 | eluz2b1 12961 | . . 3 ⊢ (2 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 1 < 2)) | |
| 4 | 1, 2, 3 | mpbir2an 711 | . 2 ⊢ 2 ∈ (ℤ≥‘2) |
| 5 | ral0 4513 | . . 3 ⊢ ∀𝑧 ∈ ∅ ¬ 𝑧 ∥ 2 | |
| 6 | fzssuz 13605 | . . . . . 6 ⊢ (2...(2 − 1)) ⊆ (ℤ≥‘2) | |
| 7 | dfss2 3969 | . . . . . 6 ⊢ ((2...(2 − 1)) ⊆ (ℤ≥‘2) ↔ ((2...(2 − 1)) ∩ (ℤ≥‘2)) = (2...(2 − 1))) | |
| 8 | 6, 7 | mpbi 230 | . . . . 5 ⊢ ((2...(2 − 1)) ∩ (ℤ≥‘2)) = (2...(2 − 1)) |
| 9 | uzdisj 13637 | . . . . 5 ⊢ ((2...(2 − 1)) ∩ (ℤ≥‘2)) = ∅ | |
| 10 | 8, 9 | eqtr3i 2767 | . . . 4 ⊢ (2...(2 − 1)) = ∅ |
| 11 | 10 | raleqi 3324 | . . 3 ⊢ (∀𝑧 ∈ (2...(2 − 1)) ¬ 𝑧 ∥ 2 ↔ ∀𝑧 ∈ ∅ ¬ 𝑧 ∥ 2) |
| 12 | 5, 11 | mpbir 231 | . 2 ⊢ ∀𝑧 ∈ (2...(2 − 1)) ¬ 𝑧 ∥ 2 |
| 13 | isprm3 16720 | . 2 ⊢ (2 ∈ ℙ ↔ (2 ∈ (ℤ≥‘2) ∧ ∀𝑧 ∈ (2...(2 − 1)) ¬ 𝑧 ∥ 2)) | |
| 14 | 4, 12, 13 | mpbir2an 711 | 1 ⊢ 2 ∈ ℙ |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 1c1 11156 < clt 11295 − cmin 11492 2c2 12321 ℤcz 12613 ℤ≥cuz 12878 ...cfz 13547 ∥ cdvds 16290 ℙcprime 16708 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-prm 16709 |
| This theorem is referenced by: 2mulprm 16730 ge2nprmge4 16738 isoddgcd1 16768 3lcm2e6 16769 pythagtriplem4 16857 pc2dvds 16917 oddprmdvds 16941 prmo2 17078 prmgaplem3 17091 lt6abl 19913 2logb9irr 26838 2logb3irr 26840 ppi2 27213 cht2 27215 1sgm2ppw 27244 perfectlem1 27273 perfectlem2 27274 perfect 27275 bpos1 27327 lgs2 27358 lgsdir2 27374 lgseisenlem2 27420 lgsquad2lem1 27428 lgsquad2lem2 27429 lgsquad3 27431 m1lgs 27432 2lgs 27451 2lgsoddprm 27460 dchrisum0flb 27554 numclwwlk5lem 30406 2sqr3minply 33791 hgt750lemd 34663 12gcd5e1 42004 fltne 42654 flt4lem5a 42662 flt4lem5b 42663 flt4lem5c 42664 flt4lem5d 42665 flt4lem5e 42666 goldbachthlem2 47533 odz2prm2pw 47550 fmtnoprmfac1 47552 fmtnoprmfac2 47554 lighneallem2 47593 lighneallem3 47594 lighneallem4 47597 proththd 47601 isodd7 47652 gcd2odd1 47655 perfectALTV 47710 7gbow 47759 sbgoldbalt 47768 sgoldbeven3prm 47770 sbgoldbo 47774 nnsum3primes4 47775 nnsum3primesle9 47781 zlmodzxznm 48414 |
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