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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 2610 | . 2 ⊢ ∅ = ∅ | |
2 | 0ex 4718 | . . 3 ⊢ ∅ ∈ V | |
3 | hasheq0 13015 | . . 3 ⊢ (∅ ∈ V → ((#‘∅) = 0 ↔ ∅ = ∅)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ((#‘∅) = 0 ↔ ∅ = ∅) |
5 | 1, 4 | mpbir 220 | 1 ⊢ (#‘∅) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ‘cfv 5804 0cc0 9815 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 |
This theorem is referenced by: hashrabrsn 13022 hashrabsn01 13023 hashrabsn1 13024 hashge0 13037 elprchashprn2 13045 hash1 13053 hashsn01 13065 hashgt12el 13070 hashgt12el2 13071 hashfzo 13076 hashfzp1 13078 hashxplem 13080 hashmap 13082 hashbc 13094 hashf1lem2 13097 hashf1 13098 hash2pwpr 13115 lsw0g 13206 ccatlid 13222 ccatrid 13223 s1nzOLD 13240 rev0 13364 repswsymballbi 13378 fsumconst 14364 incexclem 14407 incexc 14408 fprodconst 14547 sumodd 14949 hashgcdeq 15332 prmreclem4 15461 prmreclem5 15462 0hashbc 15549 ramz2 15566 cshws0 15646 psgnunilem2 17738 psgnunilem4 17740 psgn0fv0 17754 psgnsn 17763 psgnprfval1 17765 efginvrel2 17963 efgredleme 17979 efgcpbllemb 17991 frgpnabllem1 18099 gsumconst 18157 ltbwe 19293 fta1g 23731 fta1 23867 birthdaylem3 24480 ppi1 24690 musum 24717 rpvmasum 25015 umgrislfupgrlem 25788 usgraedgprv 25905 usgra1v 25919 usgrafisindb0 25937 usgrafisindb1 25938 0wlk 26075 0trl 26076 0wlkon 26077 0pth 26100 0crct 26154 0cycl 26155 0clwlk 26293 vdgr0 26427 vdgr1b 26431 vdgr1a 26433 vdusgraval 26434 rusgranumwlkl1 26474 rusgra0edg 26482 eupath 26508 f1ocnt 28946 esumcst 29452 cntmeas 29616 ballotlemfval0 29884 signsvtn0 29973 signstfvneq0 29975 signstfveq0 29980 signsvf0 29983 derangsn 30406 subfacp1lem6 30421 poimirlem25 32604 poimirlem26 32605 poimirlem27 32606 poimirlem28 32607 rp-isfinite6 36883 fzisoeu 38455 lfuhgr1v0e 40480 uvtxa01vtx 40624 vtxdg0e 40689 vtxdlfgrval 40700 rusgr1vtxlem 40787 wspn0 41131 rusgrnumwwlkl1 41172 rusgr0edg 41176 0ewlk 41282 01wlk 41284 0wlkOn 41288 0pth-av 41293 0clWlk 41298 0Crct 41300 0Cycl 41301 eupth0 41382 eulerpathpr 41408 |
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