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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 12649 | . . . 4 ⊢ 1 ∈ ℤ | |
| 2 | en2sn 9082 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 ∈ ℤ) → {𝐴} ≈ {1}) | |
| 3 | 1, 2 | mpan2 691 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ {1}) | 
| 4 | snfi 9084 | . . . 4 ⊢ {𝐴} ∈ Fin | |
| 5 | snfi 9084 | . . . 4 ⊢ {1} ∈ Fin | |
| 6 | hashen 14387 | . . . 4 ⊢ (({𝐴} ∈ Fin ∧ {1} ∈ Fin) → ((♯‘{𝐴}) = (♯‘{1}) ↔ {𝐴} ≈ {1})) | |
| 7 | 4, 5, 6 | mp2an 692 | . . 3 ⊢ ((♯‘{𝐴}) = (♯‘{1}) ↔ {𝐴} ≈ {1}) | 
| 8 | 3, 7 | sylibr 234 | . 2 ⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = (♯‘{1})) | 
| 9 | fzsn 13607 | . . . . 5 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
| 10 | 9 | fveq2d 6909 | . . . 4 ⊢ (1 ∈ ℤ → (♯‘(1...1)) = (♯‘{1})) | 
| 11 | 1nn0 12544 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
| 12 | hashfz1 14386 | . . . . 5 ⊢ (1 ∈ ℕ0 → (♯‘(1...1)) = 1) | |
| 13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ (♯‘(1...1)) = 1 | 
| 14 | 10, 13 | eqtr3di 2791 | . . 3 ⊢ (1 ∈ ℤ → (♯‘{1}) = 1) | 
| 15 | 1, 14 | ax-mp 5 | . 2 ⊢ (♯‘{1}) = 1 | 
| 16 | 8, 15 | eqtrdi 2792 | 1 ⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = 1) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 {csn 4625 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 ≈ cen 8983 Fincfn 8986 1c1 11157 ℕ0cn0 12528 ℤcz 12615 ...cfz 13548 ♯chash 14370 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-hash 14371 | 
| This theorem is referenced by: hashen1 14410 hashrabrsn 14412 hashrabsn01 14413 hashunsng 14432 hashunsngx 14433 hashprg 14435 elprchashprn2 14436 hashdifsn 14454 hashsn01 14456 hash1snb 14459 hashmap 14475 hashfun 14477 hashbclem 14492 hashbc 14493 hashf1 14497 hash2prde 14510 hash2pwpr 14516 hashge2el2dif 14520 hash7g 14526 hash3tpexb 14534 hashdifsnp1 14546 s1len 14645 ackbijnn 15865 phicl2 16806 dfphi2 16812 vdwlem8 17027 ramcl 17068 cshwshashnsame 17142 efmnd1hash 18906 symg1hash 19408 pgp0 19615 odcau 19623 sylow2a 19638 sylow3lem6 19651 prmcyg 19913 gsumsnfd 19970 ablfac1eulem 20093 ablfac1eu 20094 pgpfaclem2 20103 prmgrpsimpgd 20135 ablsimpgprmd 20136 c0snmhm 20464 0ringdif 20528 0ring01eqbi 20533 rng1nnzr 20777 fta1glem2 26209 fta1blem 26211 fta1lem 26350 vieta1lem2 26354 vieta1 26355 vmappw 27160 umgredgnlp 29165 lfuhgr1v0e 29272 usgr1vr 29273 uvtxnm1nbgr 29422 1hevtxdg1 29525 1egrvtxdg1 29528 lfgrwlkprop 29706 rusgrnumwwlkb0 29992 clwwlknon1le1 30121 eupth2eucrct 30237 fusgreghash2wspv 30355 numclwlk1lem1 30389 ex-hash 30473 0ringsubrg 33256 drngidlhash 33463 prmidl0 33479 qsidomlem1 33481 krull 33508 qsdrng 33526 rlmdim 33661 rgmoddimOLD 33662 lsatdim 33669 zarcmplem 33881 esumcst 34065 cntnevol 34230 coinflippv 34487 ccatmulgnn0dir 34558 ofcccat 34559 lpadlem2 34696 derang0 35175 poimirlem26 37654 poimirlem27 37655 poimirlem28 37656 frlmvscadiccat 42521 | 
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