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Mirrors > Home > MPE Home > Th. List > prmex | Structured version Visualization version GIF version |
Description: The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.) |
Ref | Expression |
---|---|
prmex | ⊢ ℙ ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnex 12215 | . 2 ⊢ ℕ ∈ V | |
2 | prmssnn 16610 | . 2 ⊢ ℙ ⊆ ℕ | |
3 | 1, 2 | ssexi 5322 | 1 ⊢ ℙ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3475 ℕcn 12209 ℙcprime 16605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-1cn 11165 ax-addcl 11167 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-nn 12210 df-prm 16606 |
This theorem is referenced by: 1arithlem1 16853 1arith 16857 ablfac1b 19935 vmaval 26607 sqff1o 26676 musum 26685 nnsum3primes4 46443 nnsum3primesprm 46445 nnsum3primesgbe 46447 nnsum4primesodd 46451 nnsum4primesoddALTV 46452 nnsum4primeseven 46455 nnsum4primesevenALTV 46456 |
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