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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 12000 | . . . 4 ⊢ 1 ∈ ℤ | |
2 | en2sn 8581 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 ∈ ℤ) → {𝐴} ≈ {1}) | |
3 | 1, 2 | mpan2 687 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ {1}) |
4 | snfi 8582 | . . . 4 ⊢ {𝐴} ∈ Fin | |
5 | snfi 8582 | . . . 4 ⊢ {1} ∈ Fin | |
6 | hashen 13695 | . . . 4 ⊢ (({𝐴} ∈ Fin ∧ {1} ∈ Fin) → ((♯‘{𝐴}) = (♯‘{1}) ↔ {𝐴} ≈ {1})) | |
7 | 4, 5, 6 | mp2an 688 | . . 3 ⊢ ((♯‘{𝐴}) = (♯‘{1}) ↔ {𝐴} ≈ {1}) |
8 | 3, 7 | sylibr 235 | . 2 ⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = (♯‘{1})) |
9 | 1nn0 11901 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
10 | hashfz1 13694 | . . . . 5 ⊢ (1 ∈ ℕ0 → (♯‘(1...1)) = 1) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ (♯‘(1...1)) = 1 |
12 | fzsn 12937 | . . . . 5 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
13 | 12 | fveq2d 6667 | . . . 4 ⊢ (1 ∈ ℤ → (♯‘(1...1)) = (♯‘{1})) |
14 | 11, 13 | syl5reqr 2868 | . . 3 ⊢ (1 ∈ ℤ → (♯‘{1}) = 1) |
15 | 1, 14 | ax-mp 5 | . 2 ⊢ (♯‘{1}) = 1 |
16 | 8, 15 | syl6eq 2869 | 1 ⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 {csn 4557 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ≈ cen 8494 Fincfn 8497 1c1 10526 ℕ0cn0 11885 ℤcz 11969 ...cfz 12880 ♯chash 13678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-hash 13679 |
This theorem is referenced by: hashen1 13719 hashrabrsn 13721 hashrabsn01 13722 hashunsng 13741 hashunsngx 13742 hashprg 13744 elprchashprn2 13745 hashdifsn 13763 hashsn01 13765 hash1snb 13768 hashmap 13784 hashfun 13786 hashbclem 13798 hashbc 13799 hashf1 13803 hash2prde 13816 hash2pwpr 13822 hashge2el2dif 13826 hashdifsnp1 13842 s1len 13948 ackbijnn 15171 phicl2 16093 dfphi2 16099 vdwlem8 16312 ramcl 16353 cshwshashnsame 16425 symg1hash 18452 pgp0 18650 odcau 18658 sylow2a 18673 sylow3lem6 18686 prmcyg 18943 gsumsnfd 19000 ablfac1eulem 19123 ablfac1eu 19124 pgpfaclem2 19133 prmgrpsimpgd 19165 ablsimpgprmd 19166 0ring01eqbi 19974 rng1nnzr 19975 fta1glem2 24687 fta1blem 24689 fta1lem 24823 vieta1lem2 24827 vieta1 24828 vmappw 25620 umgredgnlp 26859 lfuhgr1v0e 26963 usgr1vr 26964 uvtxnm1nbgr 27113 1hevtxdg1 27215 1egrvtxdg1 27218 lfgrwlkprop 27396 rusgrnumwwlkb0 27677 clwwlknon1le1 27807 eupth2eucrct 27923 fusgreghash2wspv 28041 numclwlk1lem1 28075 ex-hash 28159 qsidomlem1 30882 rgmoddim 30907 lsatdim 30914 esumcst 31221 cntnevol 31386 coinflippv 31640 ccatmulgnn0dir 31711 ofcccat 31712 lpadlem2 31850 derang0 32313 poimirlem26 34799 poimirlem27 34800 poimirlem28 34801 frlmvscadiccat 39023 efmnd1hash 43989 0ringdif 44069 c0snmhm 44114 |
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