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Mirrors > Home > MPE Home > Th. List > ex-hash | Structured version Visualization version GIF version |
Description: Example for df-hash 14330. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-hash | ⊢ (♯‘{0, 1, 2}) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 4637 | . . . 4 ⊢ {0, 1, 2} = ({0, 1} ∪ {2}) | |
2 | 1 | fveq2i 6905 | . . 3 ⊢ (♯‘{0, 1, 2}) = (♯‘({0, 1} ∪ {2})) |
3 | prfi 9354 | . . . 4 ⊢ {0, 1} ∈ Fin | |
4 | snfi 9075 | . . . 4 ⊢ {2} ∈ Fin | |
5 | 2ne0 12354 | . . . . . 6 ⊢ 2 ≠ 0 | |
6 | 1ne2 12458 | . . . . . . 7 ⊢ 1 ≠ 2 | |
7 | 6 | necomi 2992 | . . . . . 6 ⊢ 2 ≠ 1 |
8 | 5, 7 | nelpri 4662 | . . . . 5 ⊢ ¬ 2 ∈ {0, 1} |
9 | disjsn 4720 | . . . . 5 ⊢ (({0, 1} ∩ {2}) = ∅ ↔ ¬ 2 ∈ {0, 1}) | |
10 | 8, 9 | mpbir 230 | . . . 4 ⊢ ({0, 1} ∩ {2}) = ∅ |
11 | hashun 14381 | . . . 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 2756 | . 2 ⊢ (♯‘{0, 1, 2}) = ((♯‘{0, 1}) + (♯‘{2})) |
14 | prhash2ex 14398 | . . . 4 ⊢ (♯‘{0, 1}) = 2 | |
15 | 2z 12632 | . . . . 5 ⊢ 2 ∈ ℤ | |
16 | hashsng 14368 | . . . . 5 ⊢ (2 ∈ ℤ → (♯‘{2}) = 1) | |
17 | 15, 16 | ax-mp 5 | . . . 4 ⊢ (♯‘{2}) = 1 |
18 | 14, 17 | oveq12i 7438 | . . 3 ⊢ ((♯‘{0, 1}) + (♯‘{2})) = (2 + 1) |
19 | 2p1e3 12392 | . . 3 ⊢ (2 + 1) = 3 | |
20 | 18, 19 | eqtri 2756 | . 2 ⊢ ((♯‘{0, 1}) + (♯‘{2})) = 3 |
21 | 13, 20 | eqtri 2756 | 1 ⊢ (♯‘{0, 1, 2}) = 3 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 ∪ cun 3947 ∩ cin 3948 ∅c0 4326 {csn 4632 {cpr 4634 {ctp 4636 ‘cfv 6553 (class class class)co 7426 Fincfn 8970 0cc0 11146 1c1 11147 + caddc 11149 2c2 12305 3c3 12306 ℤcz 12596 ♯chash 14329 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-oadd 8497 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-hash 14330 |
This theorem is referenced by: (None) |
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