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Mirrors > Home > MPE Home > Th. List > hash1to3 | Structured version Visualization version GIF version |
Description: If the size of a set is between 1 and 3 (inclusively), the set is a singleton or an unordered pair or an unordered triple. (Contributed by Alexander van der Vekens, 13-Sep-2018.) |
Ref | Expression |
---|---|
hash1to3 | ⊢ ((𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3) → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashcl 14316 | . . 3 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0) | |
2 | nn01to3 12925 | . . 3 ⊢ (((♯‘𝑉) ∈ ℕ0 ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3) → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 2 ∨ (♯‘𝑉) = 3)) | |
3 | 1, 2 | syl3an1 1164 | . 2 ⊢ ((𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3) → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 2 ∨ (♯‘𝑉) = 3)) |
4 | hash1snb 14379 | . . . . . . . 8 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 1 ↔ ∃𝑎 𝑉 = {𝑎})) | |
5 | 4 | biimpa 478 | . . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 1) → ∃𝑎 𝑉 = {𝑎}) |
6 | 3mix1 1331 | . . . . . . . . . . 11 ⊢ (𝑉 = {𝑎} → (𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) | |
7 | 6 | 2eximi 1839 | . . . . . . . . . 10 ⊢ (∃𝑏∃𝑐 𝑉 = {𝑎} → ∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) |
8 | 7 | 19.23bi 2185 | . . . . . . . . 9 ⊢ (∃𝑐 𝑉 = {𝑎} → ∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) |
9 | 8 | 19.23bi 2185 | . . . . . . . 8 ⊢ (𝑉 = {𝑎} → ∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) |
10 | 9 | eximi 1838 | . . . . . . 7 ⊢ (∃𝑎 𝑉 = {𝑎} → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) |
11 | 5, 10 | syl 17 | . . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 1) → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) |
12 | 11 | expcom 415 | . . . . 5 ⊢ ((♯‘𝑉) = 1 → (𝑉 ∈ Fin → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}))) |
13 | hash2pr 14430 | . . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 2) → ∃𝑎∃𝑏 𝑉 = {𝑎, 𝑏}) | |
14 | 3mix2 1332 | . . . . . . . . . 10 ⊢ (𝑉 = {𝑎, 𝑏} → (𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) | |
15 | 14 | eximi 1838 | . . . . . . . . 9 ⊢ (∃𝑐 𝑉 = {𝑎, 𝑏} → ∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) |
16 | 15 | 19.23bi 2185 | . . . . . . . 8 ⊢ (𝑉 = {𝑎, 𝑏} → ∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) |
17 | 16 | 2eximi 1839 | . . . . . . 7 ⊢ (∃𝑎∃𝑏 𝑉 = {𝑎, 𝑏} → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) |
18 | 13, 17 | syl 17 | . . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 2) → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) |
19 | 18 | expcom 415 | . . . . 5 ⊢ ((♯‘𝑉) = 2 → (𝑉 ∈ Fin → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}))) |
20 | hash3tr 14451 | . . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 3) → ∃𝑎∃𝑏∃𝑐 𝑉 = {𝑎, 𝑏, 𝑐}) | |
21 | 3mix3 1333 | . . . . . . . . 9 ⊢ (𝑉 = {𝑎, 𝑏, 𝑐} → (𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) | |
22 | 21 | eximi 1838 | . . . . . . . 8 ⊢ (∃𝑐 𝑉 = {𝑎, 𝑏, 𝑐} → ∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) |
23 | 22 | 2eximi 1839 | . . . . . . 7 ⊢ (∃𝑎∃𝑏∃𝑐 𝑉 = {𝑎, 𝑏, 𝑐} → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) |
24 | 20, 23 | syl 17 | . . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 3) → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) |
25 | 24 | expcom 415 | . . . . 5 ⊢ ((♯‘𝑉) = 3 → (𝑉 ∈ Fin → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}))) |
26 | 12, 19, 25 | 3jaoi 1428 | . . . 4 ⊢ (((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 2 ∨ (♯‘𝑉) = 3) → (𝑉 ∈ Fin → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}))) |
27 | 26 | com12 32 | . . 3 ⊢ (𝑉 ∈ Fin → (((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 2 ∨ (♯‘𝑉) = 3) → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}))) |
28 | 27 | 3ad2ant1 1134 | . 2 ⊢ ((𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3) → (((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 2 ∨ (♯‘𝑉) = 3) → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}))) |
29 | 3, 28 | mpd 15 | 1 ⊢ ((𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3) → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∨ w3o 1087 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {csn 4629 {cpr 4631 {ctp 4633 class class class wbr 5149 ‘cfv 6544 Fincfn 8939 1c1 11111 ≤ cle 11249 2c2 12267 3c3 12268 ℕ0cn0 12472 ♯chash 14290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-3o 8468 df-oadd 8470 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-hash 14291 |
This theorem is referenced by: friendship 29652 |
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