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Mirrors > Home > MPE Home > Th. List > hash0 | Structured version Visualization version GIF version |
Description: The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.) |
Ref | Expression |
---|---|
hash0 | ⊢ (♯‘∅) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . 2 ⊢ ∅ = ∅ | |
2 | 0ex 5312 | . . 3 ⊢ ∅ ∈ V | |
3 | hasheq0 14380 | . . 3 ⊢ (∅ ∈ V → ((♯‘∅) = 0 ↔ ∅ = ∅)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ((♯‘∅) = 0 ↔ ∅ = ∅) |
5 | 1, 4 | mpbir 230 | 1 ⊢ (♯‘∅) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∅c0 4325 ‘cfv 6554 0cc0 11158 ♯chash 14347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 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 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-hash 14348 |
This theorem is referenced by: hashrabrsn 14389 hashrabsn01 14390 hashrabsn1 14391 hashge0 14404 elprchashprn2 14413 hash1 14421 hashsn01 14433 hashgt12el 14439 hashgt12el2 14440 hashfzo 14446 hashfzp1 14448 hashxplem 14450 hashmap 14452 hashbc 14470 hashf1lem2 14475 hashf1 14476 hash2pwpr 14495 wrdnfi 14556 lsw0g 14574 ccatlid 14594 ccatrid 14595 rev0 14772 repswsymballbi 14788 fsumconst 15794 incexclem 15840 incexc 15841 fprodconst 15980 sumodd 16390 hashgcdeq 16791 prmreclem4 16921 prmreclem5 16922 0hashbc 17009 ramz2 17026 cshws0 17104 psgnunilem2 19493 psgnunilem4 19495 psgn0fv0 19509 psgnsn 19518 psgnprfval1 19520 efginvrel2 19725 efgredleme 19741 efgcpbllemb 19753 frgpnabllem1 19871 gsumconst 19932 ltbwe 22051 fta1g 26197 fta1 26336 birthdaylem3 26981 ppi1 27192 musum 27219 rpvmasum 27555 umgrislfupgrlem 29058 lfuhgr1v0e 29190 vtxdg0e 29411 vtxdlfgrval 29422 rusgr1vtxlem 29524 wspn0 29858 rusgrnumwwlkl1 29902 rusgr0edg 29907 clwwlknonel 30028 clwwlknon1le1 30034 0ewlk 30047 0wlk 30049 0wlkon 30053 0pth 30058 0clwlk 30063 0crct 30066 0cycl 30067 eupth0 30147 eulerpathpr 30173 wlkl0 30300 f1ocnt 32704 hashxpe 32711 chnub 32881 1arithidom 33412 lvecdim0 33501 fldext2chn 33606 esumcst 33896 cntmeas 34059 ballotlemfval0 34329 signsvtn0 34416 signstfvneq0 34418 signstfveq0 34423 signsvf0 34426 lpadright 34530 derangsn 34998 subfacp1lem6 35013 poimirlem25 37346 poimirlem26 37347 poimirlem27 37348 poimirlem28 37349 rp-isfinite6 43185 fzisoeu 44915 upwordnul 46499 |
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