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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 108 | . . 3 ⊢ ((𝑀 ∈ Fin ∧ (♯‘𝑀) < 0) → (♯‘𝑀) < 0) | |
2 | hashcl 10024 | . . . . 5 ⊢ (𝑀 ∈ Fin → (♯‘𝑀) ∈ ℕ0) | |
3 | nn0nlt0 8591 | . . . . 5 ⊢ ((♯‘𝑀) ∈ ℕ0 → ¬ (♯‘𝑀) < 0) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝑀 ∈ Fin → ¬ (♯‘𝑀) < 0) |
5 | 4 | adantr 270 | . . 3 ⊢ ((𝑀 ∈ Fin ∧ (♯‘𝑀) < 0) → ¬ (♯‘𝑀) < 0) |
6 | 1, 5 | pm2.21dd 583 | . 2 ⊢ ((𝑀 ∈ Fin ∧ (♯‘𝑀) < 0) → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
7 | orc 666 | . . . 4 ⊢ (((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1) → (((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1) ∨ 1 < (♯‘𝑀))) | |
8 | fz01or 9418 | . . . 4 ⊢ ((♯‘𝑀) ∈ (0...1) ↔ ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1)) | |
9 | df-3or 921 | . . . 4 ⊢ (((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀)) ↔ (((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1) ∨ 1 < (♯‘𝑀))) | |
10 | 7, 8, 9 | 3imtr4i 199 | . . 3 ⊢ ((♯‘𝑀) ∈ (0...1) → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
11 | 10 | adantl 271 | . 2 ⊢ ((𝑀 ∈ Fin ∧ (♯‘𝑀) ∈ (0...1)) → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
12 | 3mix3 1110 | . . 3 ⊢ (1 < (♯‘𝑀) → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) | |
13 | 12 | adantl 271 | . 2 ⊢ ((𝑀 ∈ Fin ∧ 1 < (♯‘𝑀)) → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
14 | 2 | nn0zd 8762 | . . 3 ⊢ (𝑀 ∈ Fin → (♯‘𝑀) ∈ ℤ) |
15 | 0zd 8658 | . . 3 ⊢ (𝑀 ∈ Fin → 0 ∈ ℤ) | |
16 | 1zzd 8673 | . . 3 ⊢ (𝑀 ∈ Fin → 1 ∈ ℤ) | |
17 | fztri3or 9348 | . . 3 ⊢ (((♯‘𝑀) ∈ ℤ ∧ 0 ∈ ℤ ∧ 1 ∈ ℤ) → ((♯‘𝑀) < 0 ∨ (♯‘𝑀) ∈ (0...1) ∨ 1 < (♯‘𝑀))) | |
18 | 14, 15, 16, 17 | syl3anc 1170 | . 2 ⊢ (𝑀 ∈ Fin → ((♯‘𝑀) < 0 ∨ (♯‘𝑀) ∈ (0...1) ∨ 1 < (♯‘𝑀))) |
19 | 6, 11, 13, 18 | mpjao3dan 1239 | 1 ⊢ (𝑀 ∈ Fin → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ∨ wo 662 ∨ w3o 919 = wceq 1285 ∈ wcel 1434 class class class wbr 3811 ‘cfv 4969 (class class class)co 5591 Fincfn 6387 0cc0 7253 1c1 7254 < clt 7425 ℕ0cn0 8565 ℤcz 8646 ...cfz 9319 ♯chash 10018 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-iinf 4366 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-addcom 7348 ax-addass 7350 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-0id 7356 ax-rnegex 7357 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-apti 7363 ax-pre-ltadd 7364 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4084 df-iord 4157 df-on 4159 df-ilim 4160 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-recs 6002 df-frec 6088 df-er 6222 df-en 6388 df-dom 6389 df-fin 6390 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-inn 8317 df-n0 8566 df-z 8647 df-uz 8915 df-fz 9320 df-ihash 10019 |
This theorem is referenced by: (None) |
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