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Mirrors > Home > ILE Home > Th. List > hash0 | 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 2170 | . 2 ⊢ ∅ = ∅ | |
2 | 0fin 6858 | . . 3 ⊢ ∅ ∈ Fin | |
3 | fihasheq0 10715 | . . 3 ⊢ (∅ ∈ Fin → ((♯‘∅) = 0 ↔ ∅ = ∅)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ((♯‘∅) = 0 ↔ ∅ = ∅) |
5 | 1, 4 | mpbir 145 | 1 ⊢ (♯‘∅) = 0 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1348 ∈ wcel 2141 ∅c0 3414 ‘cfv 5196 Fincfn 6714 0cc0 7761 ♯chash 10696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-recs 6281 df-frec 6367 df-1o 6392 df-er 6509 df-en 6715 df-dom 6716 df-fin 6717 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-inn 8866 df-n0 9123 df-z 9200 df-uz 9475 df-fz 9953 df-ihash 10697 |
This theorem is referenced by: hash1 10733 hashfzo 10744 hashfzp1 10746 hashxp 10748 zfz1iso 10763 fsumconst 11404 fprodconst 11570 hashgcdeq 12180 |
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