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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 12540 | . . . 4 ⊢ 1 ∈ ℤ | |
2 | en2sn 8992 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 ∈ ℤ) → {𝐴} ≈ {1}) | |
3 | 1, 2 | mpan2 690 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ {1}) |
4 | snfi 8995 | . . . 4 ⊢ {𝐴} ∈ Fin | |
5 | snfi 8995 | . . . 4 ⊢ {1} ∈ Fin | |
6 | hashen 14254 | . . . 4 ⊢ (({𝐴} ∈ Fin ∧ {1} ∈ Fin) → ((♯‘{𝐴}) = (♯‘{1}) ↔ {𝐴} ≈ {1})) | |
7 | 4, 5, 6 | mp2an 691 | . . 3 ⊢ ((♯‘{𝐴}) = (♯‘{1}) ↔ {𝐴} ≈ {1}) |
8 | 3, 7 | sylibr 233 | . 2 ⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = (♯‘{1})) |
9 | fzsn 13490 | . . . . 5 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
10 | 9 | fveq2d 6851 | . . . 4 ⊢ (1 ∈ ℤ → (♯‘(1...1)) = (♯‘{1})) |
11 | 1nn0 12436 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
12 | hashfz1 14253 | . . . . 5 ⊢ (1 ∈ ℕ0 → (♯‘(1...1)) = 1) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ (♯‘(1...1)) = 1 |
14 | 10, 13 | eqtr3di 2792 | . . 3 ⊢ (1 ∈ ℤ → (♯‘{1}) = 1) |
15 | 1, 14 | ax-mp 5 | . 2 ⊢ (♯‘{1}) = 1 |
16 | 8, 15 | eqtrdi 2793 | 1 ⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 {csn 4591 class class class wbr 5110 ‘cfv 6501 (class class class)co 7362 ≈ cen 8887 Fincfn 8890 1c1 11059 ℕ0cn0 12420 ℤcz 12506 ...cfz 13431 ♯chash 14237 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-hash 14238 |
This theorem is referenced by: hashen1 14277 hashrabrsn 14279 hashrabsn01 14280 hashunsng 14299 hashunsngx 14300 hashprg 14302 elprchashprn2 14303 hashdifsn 14321 hashsn01 14323 hash1snb 14326 hashmap 14342 hashfun 14344 hashbclem 14356 hashbc 14357 hashf1 14363 hash2prde 14376 hash2pwpr 14382 hashge2el2dif 14386 hashdifsnp1 14402 s1len 14501 ackbijnn 15720 phicl2 16647 dfphi2 16653 vdwlem8 16867 ramcl 16908 cshwshashnsame 16983 efmnd1hash 18709 symg1hash 19178 pgp0 19385 odcau 19393 sylow2a 19408 sylow3lem6 19421 prmcyg 19678 gsumsnfd 19735 ablfac1eulem 19858 ablfac1eu 19859 pgpfaclem2 19868 prmgrpsimpgd 19900 ablsimpgprmd 19901 0ring01eqbi 20759 rng1nnzr 20760 fta1glem2 25547 fta1blem 25549 fta1lem 25683 vieta1lem2 25687 vieta1 25688 vmappw 26481 umgredgnlp 28140 lfuhgr1v0e 28244 usgr1vr 28245 uvtxnm1nbgr 28394 1hevtxdg1 28496 1egrvtxdg1 28499 lfgrwlkprop 28677 rusgrnumwwlkb0 28958 clwwlknon1le1 29087 eupth2eucrct 29203 fusgreghash2wspv 29321 numclwlk1lem1 29355 ex-hash 29439 0ringsubrg 32106 prmidl0 32263 qsidomlem1 32265 krull 32280 rgmoddim 32347 lsatdim 32354 zarcmplem 32502 esumcst 32702 cntnevol 32867 coinflippv 33123 ccatmulgnn0dir 33194 ofcccat 33195 lpadlem2 33333 derang0 33803 poimirlem26 36133 poimirlem27 36134 poimirlem28 36135 frlmvscadiccat 40710 0ringdif 46242 c0snmhm 46287 |
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