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Mirrors > Home > MPE Home > Th. List > hashsng | Structured version Visualization version GIF version |
Description: The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.) |
Ref | Expression |
---|---|
hashsng | ⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 12630 | . . . 4 ⊢ 1 ∈ ℤ | |
2 | en2sn 9072 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 ∈ ℤ) → {𝐴} ≈ {1}) | |
3 | 1, 2 | mpan2 689 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ {1}) |
4 | snfi 9075 | . . . 4 ⊢ {𝐴} ∈ Fin | |
5 | snfi 9075 | . . . 4 ⊢ {1} ∈ Fin | |
6 | hashen 14346 | . . . 4 ⊢ (({𝐴} ∈ Fin ∧ {1} ∈ Fin) → ((♯‘{𝐴}) = (♯‘{1}) ↔ {𝐴} ≈ {1})) | |
7 | 4, 5, 6 | mp2an 690 | . . 3 ⊢ ((♯‘{𝐴}) = (♯‘{1}) ↔ {𝐴} ≈ {1}) |
8 | 3, 7 | sylibr 233 | . 2 ⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = (♯‘{1})) |
9 | fzsn 13583 | . . . . 5 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
10 | 9 | fveq2d 6906 | . . . 4 ⊢ (1 ∈ ℤ → (♯‘(1...1)) = (♯‘{1})) |
11 | 1nn0 12526 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
12 | hashfz1 14345 | . . . . 5 ⊢ (1 ∈ ℕ0 → (♯‘(1...1)) = 1) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ (♯‘(1...1)) = 1 |
14 | 10, 13 | eqtr3di 2783 | . . 3 ⊢ (1 ∈ ℤ → (♯‘{1}) = 1) |
15 | 1, 14 | ax-mp 5 | . 2 ⊢ (♯‘{1}) = 1 |
16 | 8, 15 | eqtrdi 2784 | 1 ⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 {csn 4632 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 ≈ cen 8967 Fincfn 8970 1c1 11147 ℕ0cn0 12510 ℤcz 12596 ...cfz 13524 ♯chash 14329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-hash 14330 |
This theorem is referenced by: hashen1 14369 hashrabrsn 14371 hashrabsn01 14372 hashunsng 14391 hashunsngx 14392 hashprg 14394 elprchashprn2 14395 hashdifsn 14413 hashsn01 14415 hash1snb 14418 hashmap 14434 hashfun 14436 hashbclem 14451 hashbc 14452 hashf1 14458 hash2prde 14471 hash2pwpr 14477 hashge2el2dif 14481 hashdifsnp1 14497 s1len 14596 ackbijnn 15814 phicl2 16744 dfphi2 16750 vdwlem8 16964 ramcl 17005 cshwshashnsame 17080 efmnd1hash 18851 symg1hash 19351 pgp0 19558 odcau 19566 sylow2a 19581 sylow3lem6 19594 prmcyg 19856 gsumsnfd 19913 ablfac1eulem 20036 ablfac1eu 20037 pgpfaclem2 20046 prmgrpsimpgd 20078 ablsimpgprmd 20079 c0snmhm 20409 0ringdif 20471 0ring01eqbi 20476 rng1nnzr 20670 fta1glem2 26123 fta1blem 26125 fta1lem 26262 vieta1lem2 26266 vieta1 26267 vmappw 27068 umgredgnlp 28980 lfuhgr1v0e 29087 usgr1vr 29088 uvtxnm1nbgr 29237 1hevtxdg1 29340 1egrvtxdg1 29343 lfgrwlkprop 29521 rusgrnumwwlkb0 29802 clwwlknon1le1 29931 eupth2eucrct 30047 fusgreghash2wspv 30165 numclwlk1lem1 30199 ex-hash 30283 0ringsubrg 32969 drngidlhash 33175 prmidl0 33191 qsidomlem1 33193 krull 33216 qsdrng 33233 rlmdim 33340 rgmoddimOLD 33341 lsatdim 33348 zarcmplem 33515 esumcst 33715 cntnevol 33880 coinflippv 34136 ccatmulgnn0dir 34207 ofcccat 34208 lpadlem2 34345 derang0 34812 poimirlem26 37152 poimirlem27 37153 poimirlem28 37154 frlmvscadiccat 41777 |
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