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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 2780 | . 2 ⊢ ∅ = ∅ | |
2 | 0ex 5072 | . . 3 ⊢ ∅ ∈ V | |
3 | hasheq0 13545 | . . 3 ⊢ (∅ ∈ V → ((♯‘∅) = 0 ↔ ∅ = ∅)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ((♯‘∅) = 0 ↔ ∅ = ∅) |
5 | 1, 4 | mpbir 223 | 1 ⊢ (♯‘∅) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1508 ∈ wcel 2051 Vcvv 3417 ∅c0 4181 ‘cfv 6193 0cc0 10341 ♯chash 13511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-int 4755 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-om 7403 df-1st 7507 df-2nd 7508 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-1o 7911 df-er 8095 df-en 8313 df-dom 8314 df-sdom 8315 df-fin 8316 df-card 9168 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-nn 11446 df-n0 11714 df-z 11800 df-uz 12065 df-fz 12715 df-hash 13512 |
This theorem is referenced by: hashrabrsn 13552 hashrabsn01 13553 hashrabsn1 13554 hashge0 13567 elprchashprn2 13576 hash1 13584 hashsn01 13596 hashgt12el 13602 hashgt12el2 13603 hashfzo 13609 hashfzp1 13611 hashxplem 13613 hashmap 13615 hashbc 13630 hashf1lem2 13633 hashf1 13634 hash2pwpr 13651 wrdnfi 13716 lsw0g 13735 ccatlid 13755 ccatrid 13756 rev0 13989 repswsymballbi 14005 fsumconst 15011 incexclem 15057 incexc 15058 fprodconst 15198 sumodd 15605 hashgcdeq 15988 prmreclem4 16117 prmreclem5 16118 0hashbc 16205 ramz2 16222 cshws0 16297 psgnunilem2 18397 psgnunilem4 18399 psgn0fv0 18413 psgnsn 18422 psgnprfval1 18424 efginvrel2 18623 efgredleme 18640 efgcpbllemb 18653 frgpnabllem1 18761 gsumconst 18819 ltbwe 19978 fta1g 24479 fta1 24615 birthdaylem3 25248 ppi1 25458 musum 25485 rpvmasum 25819 umgrislfupgrlem 26625 lfuhgr1v0e 26754 vtxdg0e 26974 vtxdlfgrval 26985 rusgr1vtxlem 27087 wspn0 27445 rusgrnumwwlkl1 27489 rusgr0edg 27494 clwwlknonel 27638 clwwlknon1le1 27644 0ewlk 27658 0wlk 27660 0wlkon 27664 0pth 27669 0clwlk 27674 0crct 27677 0cycl 27678 eupth0 27758 eulerpathpr 27785 wlkl0 27935 f1ocnt 30296 hashxpe 30300 lvecdim0 30666 esumcst 30998 cntmeas 31162 ballotlemfval0 31431 signsvtn0 31518 signsvtn0OLD 31519 signstfvneq0 31521 signstfveq0 31526 signstfveq0OLD 31527 signsvf0 31530 lpadright 31635 derangsn 32042 subfacp1lem6 32057 poimirlem25 34398 poimirlem26 34399 poimirlem27 34400 poimirlem28 34401 rp-isfinite6 39321 fzisoeu 41031 |
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