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Mirrors > Home > ILE Home > Th. List > hashfiv01gt1 | GIF version |
Description: The size of a finite set is either 0 or 1 or greater than 1. (Contributed by Jim Kingdon, 21-Feb-2022.) |
Ref | Expression |
---|---|
hashfiv01gt1 | ⊢ (𝑀 ∈ Fin → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . 3 ⊢ ((𝑀 ∈ Fin ∧ (♯‘𝑀) < 0) → (♯‘𝑀) < 0) | |
2 | hashcl 10304 | . . . . 5 ⊢ (𝑀 ∈ Fin → (♯‘𝑀) ∈ ℕ0) | |
3 | nn0nlt0 8797 | . . . . 5 ⊢ ((♯‘𝑀) ∈ ℕ0 → ¬ (♯‘𝑀) < 0) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝑀 ∈ Fin → ¬ (♯‘𝑀) < 0) |
5 | 4 | adantr 271 | . . 3 ⊢ ((𝑀 ∈ Fin ∧ (♯‘𝑀) < 0) → ¬ (♯‘𝑀) < 0) |
6 | 1, 5 | pm2.21dd 588 | . 2 ⊢ ((𝑀 ∈ Fin ∧ (♯‘𝑀) < 0) → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
7 | orc 671 | . . . 4 ⊢ (((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1) → (((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1) ∨ 1 < (♯‘𝑀))) | |
8 | fz01or 9674 | . . . 4 ⊢ ((♯‘𝑀) ∈ (0...1) ↔ ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1)) | |
9 | df-3or 928 | . . . 4 ⊢ (((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀)) ↔ (((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1) ∨ 1 < (♯‘𝑀))) | |
10 | 7, 8, 9 | 3imtr4i 200 | . . 3 ⊢ ((♯‘𝑀) ∈ (0...1) → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
11 | 10 | adantl 272 | . 2 ⊢ ((𝑀 ∈ Fin ∧ (♯‘𝑀) ∈ (0...1)) → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
12 | 3mix3 1117 | . . 3 ⊢ (1 < (♯‘𝑀) → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) | |
13 | 12 | adantl 272 | . 2 ⊢ ((𝑀 ∈ Fin ∧ 1 < (♯‘𝑀)) → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
14 | 2 | nn0zd 8965 | . . 3 ⊢ (𝑀 ∈ Fin → (♯‘𝑀) ∈ ℤ) |
15 | 0zd 8860 | . . 3 ⊢ (𝑀 ∈ Fin → 0 ∈ ℤ) | |
16 | 1zzd 8875 | . . 3 ⊢ (𝑀 ∈ Fin → 1 ∈ ℤ) | |
17 | fztri3or 9602 | . . 3 ⊢ (((♯‘𝑀) ∈ ℤ ∧ 0 ∈ ℤ ∧ 1 ∈ ℤ) → ((♯‘𝑀) < 0 ∨ (♯‘𝑀) ∈ (0...1) ∨ 1 < (♯‘𝑀))) | |
18 | 14, 15, 16, 17 | syl3anc 1181 | . 2 ⊢ (𝑀 ∈ Fin → ((♯‘𝑀) < 0 ∨ (♯‘𝑀) ∈ (0...1) ∨ 1 < (♯‘𝑀))) |
19 | 6, 11, 13, 18 | mpjao3dan 1250 | 1 ⊢ (𝑀 ∈ Fin → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 667 ∨ w3o 926 = wceq 1296 ∈ wcel 1445 class class class wbr 3867 ‘cfv 5049 (class class class)co 5690 Fincfn 6537 0cc0 7447 1c1 7448 < clt 7619 ℕ0cn0 8771 ℤcz 8848 ...cfz 9573 ♯chash 10298 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-addass 7544 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-iord 4217 df-on 4219 df-ilim 4220 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-recs 6108 df-frec 6194 df-er 6332 df-en 6538 df-dom 6539 df-fin 6540 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-inn 8521 df-n0 8772 df-z 8849 df-uz 9119 df-fz 9574 df-ihash 10299 |
This theorem is referenced by: (None) |
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