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| Description: Example for df-hash 14370. (Contributed by AV, 4-Sep-2021.) | 
| Ref | Expression | 
|---|---|
| ex-hash | ⊢ (♯‘{0, 1, 2}) = 3 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-tp 4631 | . . . 4 ⊢ {0, 1, 2} = ({0, 1} ∪ {2}) | |
| 2 | 1 | fveq2i 6909 | . . 3 ⊢ (♯‘{0, 1, 2}) = (♯‘({0, 1} ∪ {2})) | 
| 3 | prfi 9363 | . . . 4 ⊢ {0, 1} ∈ Fin | |
| 4 | snfi 9083 | . . . 4 ⊢ {2} ∈ Fin | |
| 5 | 2ne0 12370 | . . . . . 6 ⊢ 2 ≠ 0 | |
| 6 | 1ne2 12474 | . . . . . . 7 ⊢ 1 ≠ 2 | |
| 7 | 6 | necomi 2995 | . . . . . 6 ⊢ 2 ≠ 1 | 
| 8 | 5, 7 | nelpri 4655 | . . . . 5 ⊢ ¬ 2 ∈ {0, 1} | 
| 9 | disjsn 4711 | . . . . 5 ⊢ (({0, 1} ∩ {2}) = ∅ ↔ ¬ 2 ∈ {0, 1}) | |
| 10 | 8, 9 | mpbir 231 | . . . 4 ⊢ ({0, 1} ∩ {2}) = ∅ | 
| 11 | hashun 14421 | . . . 4 ⊢ (({0, 1} ∈ Fin ∧ {2} ∈ Fin ∧ ({0, 1} ∩ {2}) = ∅) → (♯‘({0, 1} ∪ {2})) = ((♯‘{0, 1}) + (♯‘{2}))) | |
| 12 | 3, 4, 10, 11 | mp3an 1463 | . . 3 ⊢ (♯‘({0, 1} ∪ {2})) = ((♯‘{0, 1}) + (♯‘{2})) | 
| 13 | 2, 12 | eqtri 2765 | . 2 ⊢ (♯‘{0, 1, 2}) = ((♯‘{0, 1}) + (♯‘{2})) | 
| 14 | prhash2ex 14438 | . . . 4 ⊢ (♯‘{0, 1}) = 2 | |
| 15 | 2z 12649 | . . . . 5 ⊢ 2 ∈ ℤ | |
| 16 | hashsng 14408 | . . . . 5 ⊢ (2 ∈ ℤ → (♯‘{2}) = 1) | |
| 17 | 15, 16 | ax-mp 5 | . . . 4 ⊢ (♯‘{2}) = 1 | 
| 18 | 14, 17 | oveq12i 7443 | . . 3 ⊢ ((♯‘{0, 1}) + (♯‘{2})) = (2 + 1) | 
| 19 | 2p1e3 12408 | . . 3 ⊢ (2 + 1) = 3 | |
| 20 | 18, 19 | eqtri 2765 | . 2 ⊢ ((♯‘{0, 1}) + (♯‘{2})) = 3 | 
| 21 | 13, 20 | eqtri 2765 | 1 ⊢ (♯‘{0, 1, 2}) = 3 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 {csn 4626 {cpr 4628 {ctp 4630 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 0cc0 11155 1c1 11156 + caddc 11158 2c2 12321 3c3 12322 ℤcz 12613 ♯chash 14369 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 | 
| This theorem is referenced by: usgrexmpl1tri 47984 | 
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