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Mirrors > Home > MPE Home > Th. List > ex-hash | Structured version Visualization version GIF version |
Description: Example for df-hash 13201. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-hash | ⊢ (♯‘{0, 1, 2}) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 4258 | . . . 4 ⊢ {0, 1, 2} = ({0, 1} ∪ {2}) | |
2 | 1 | fveq2i 6275 | . . 3 ⊢ (♯‘{0, 1, 2}) = (♯‘({0, 1} ∪ {2})) |
3 | prfi 8319 | . . . 4 ⊢ {0, 1} ∈ Fin | |
4 | snfi 8122 | . . . 4 ⊢ {2} ∈ Fin | |
5 | 2ne0 11194 | . . . . . 6 ⊢ 2 ≠ 0 | |
6 | 1ne2 11321 | . . . . . . 7 ⊢ 1 ≠ 2 | |
7 | 6 | necomi 2918 | . . . . . 6 ⊢ 2 ≠ 1 |
8 | 5, 7 | nelpri 4277 | . . . . 5 ⊢ ¬ 2 ∈ {0, 1} |
9 | disjsn 4321 | . . . . 5 ⊢ (({0, 1} ∩ {2}) = ∅ ↔ ¬ 2 ∈ {0, 1}) | |
10 | 8, 9 | mpbir 221 | . . . 4 ⊢ ({0, 1} ∩ {2}) = ∅ |
11 | hashun 13252 | . . . 4 ⊢ (({0, 1} ∈ Fin ∧ {2} ∈ Fin ∧ ({0, 1} ∩ {2}) = ∅) → (♯‘({0, 1} ∪ {2})) = ((♯‘{0, 1}) + (♯‘{2}))) | |
12 | 3, 4, 10, 11 | mp3an 1505 | . . 3 ⊢ (♯‘({0, 1} ∪ {2})) = ((♯‘{0, 1}) + (♯‘{2})) |
13 | 2, 12 | eqtri 2714 | . 2 ⊢ (♯‘{0, 1, 2}) = ((♯‘{0, 1}) + (♯‘{2})) |
14 | prhash2ex 13268 | . . . 4 ⊢ (♯‘{0, 1}) = 2 | |
15 | 2z 11490 | . . . . 5 ⊢ 2 ∈ ℤ | |
16 | hashsng 13240 | . . . . 5 ⊢ (2 ∈ ℤ → (♯‘{2}) = 1) | |
17 | 15, 16 | ax-mp 5 | . . . 4 ⊢ (♯‘{2}) = 1 |
18 | 14, 17 | oveq12i 6745 | . . 3 ⊢ ((♯‘{0, 1}) + (♯‘{2})) = (2 + 1) |
19 | 2p1e3 11232 | . . 3 ⊢ (2 + 1) = 3 | |
20 | 18, 19 | eqtri 2714 | . 2 ⊢ ((♯‘{0, 1}) + (♯‘{2})) = 3 |
21 | 13, 20 | eqtri 2714 | 1 ⊢ (♯‘{0, 1, 2}) = 3 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1564 ∈ wcel 2071 ∪ cun 3646 ∩ cin 3647 ∅c0 3991 {csn 4253 {cpr 4255 {ctp 4257 ‘cfv 5969 (class class class)co 6733 Fincfn 8040 0cc0 10017 1c1 10018 + caddc 10020 2c2 11151 3c3 11152 ℤcz 11458 ♯chash 13200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-8 2073 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-rep 4847 ax-sep 4857 ax-nul 4865 ax-pow 4916 ax-pr 4979 ax-un 7034 ax-cnex 10073 ax-resscn 10074 ax-1cn 10075 ax-icn 10076 ax-addcl 10077 ax-addrcl 10078 ax-mulcl 10079 ax-mulrcl 10080 ax-mulcom 10081 ax-addass 10082 ax-mulass 10083 ax-distr 10084 ax-i2m1 10085 ax-1ne0 10086 ax-1rid 10087 ax-rnegex 10088 ax-rrecex 10089 ax-cnre 10090 ax-pre-lttri 10091 ax-pre-lttrn 10092 ax-pre-ltadd 10093 ax-pre-mulgt0 10094 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1567 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-nel 2968 df-ral 2987 df-rex 2988 df-reu 2989 df-rmo 2990 df-rab 2991 df-v 3274 df-sbc 3510 df-csb 3608 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-pss 3664 df-nul 3992 df-if 4163 df-pw 4236 df-sn 4254 df-pr 4256 df-tp 4258 df-op 4260 df-uni 4513 df-int 4552 df-iun 4598 df-br 4729 df-opab 4789 df-mpt 4806 df-tr 4829 df-id 5096 df-eprel 5101 df-po 5107 df-so 5108 df-fr 5145 df-we 5147 df-xp 5192 df-rel 5193 df-cnv 5194 df-co 5195 df-dm 5196 df-rn 5197 df-res 5198 df-ima 5199 df-pred 5761 df-ord 5807 df-on 5808 df-lim 5809 df-suc 5810 df-iota 5932 df-fun 5971 df-fn 5972 df-f 5973 df-f1 5974 df-fo 5975 df-f1o 5976 df-fv 5977 df-riota 6694 df-ov 6736 df-oprab 6737 df-mpt2 6738 df-om 7151 df-1st 7253 df-2nd 7254 df-wrecs 7495 df-recs 7556 df-rdg 7594 df-1o 7648 df-oadd 7652 df-er 7830 df-en 8041 df-dom 8042 df-sdom 8043 df-fin 8044 df-card 8846 df-cda 9071 df-pnf 10157 df-mnf 10158 df-xr 10159 df-ltxr 10160 df-le 10161 df-sub 10349 df-neg 10350 df-nn 11102 df-2 11160 df-3 11161 df-n0 11374 df-z 11459 df-uz 11769 df-fz 12409 df-hash 13201 |
This theorem is referenced by: (None) |
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