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Mirrors > Home > MPE Home > Th. List > ex-hash | Structured version Visualization version GIF version |
Description: Example for df-hash 14331. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-hash | ⊢ (♯‘{0, 1, 2}) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 4635 | . . . 4 ⊢ {0, 1, 2} = ({0, 1} ∪ {2}) | |
2 | 1 | fveq2i 6899 | . . 3 ⊢ (♯‘{0, 1, 2}) = (♯‘({0, 1} ∪ {2})) |
3 | prfi 9351 | . . . 4 ⊢ {0, 1} ∈ Fin | |
4 | snfi 9071 | . . . 4 ⊢ {2} ∈ Fin | |
5 | 2ne0 12354 | . . . . . 6 ⊢ 2 ≠ 0 | |
6 | 1ne2 12458 | . . . . . . 7 ⊢ 1 ≠ 2 | |
7 | 6 | necomi 2984 | . . . . . 6 ⊢ 2 ≠ 1 |
8 | 5, 7 | nelpri 4659 | . . . . 5 ⊢ ¬ 2 ∈ {0, 1} |
9 | disjsn 4717 | . . . . 5 ⊢ (({0, 1} ∩ {2}) = ∅ ↔ ¬ 2 ∈ {0, 1}) | |
10 | 8, 9 | mpbir 230 | . . . 4 ⊢ ({0, 1} ∩ {2}) = ∅ |
11 | hashun 14382 | . . . 4 ⊢ (({0, 1} ∈ Fin ∧ {2} ∈ Fin ∧ ({0, 1} ∩ {2}) = ∅) → (♯‘({0, 1} ∪ {2})) = ((♯‘{0, 1}) + (♯‘{2}))) | |
12 | 3, 4, 10, 11 | mp3an 1457 | . . 3 ⊢ (♯‘({0, 1} ∪ {2})) = ((♯‘{0, 1}) + (♯‘{2})) |
13 | 2, 12 | eqtri 2753 | . 2 ⊢ (♯‘{0, 1, 2}) = ((♯‘{0, 1}) + (♯‘{2})) |
14 | prhash2ex 14399 | . . . 4 ⊢ (♯‘{0, 1}) = 2 | |
15 | 2z 12632 | . . . . 5 ⊢ 2 ∈ ℤ | |
16 | hashsng 14369 | . . . . 5 ⊢ (2 ∈ ℤ → (♯‘{2}) = 1) | |
17 | 15, 16 | ax-mp 5 | . . . 4 ⊢ (♯‘{2}) = 1 |
18 | 14, 17 | oveq12i 7431 | . . 3 ⊢ ((♯‘{0, 1}) + (♯‘{2})) = (2 + 1) |
19 | 2p1e3 12392 | . . 3 ⊢ (2 + 1) = 3 | |
20 | 18, 19 | eqtri 2753 | . 2 ⊢ ((♯‘{0, 1}) + (♯‘{2})) = 3 |
21 | 13, 20 | eqtri 2753 | 1 ⊢ (♯‘{0, 1, 2}) = 3 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 ∪ cun 3942 ∩ cin 3943 ∅c0 4322 {csn 4630 {cpr 4632 {ctp 4634 ‘cfv 6549 (class class class)co 7419 Fincfn 8964 0cc0 11145 1c1 11146 + caddc 11148 2c2 12305 3c3 12306 ℤcz 12596 ♯chash 14330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9931 df-card 9969 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-hash 14331 |
This theorem is referenced by: (None) |
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