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Mirrors > Home > ILE Home > Th. List > fihasheq0 | GIF version |
Description: Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.) (Intuitionized by Jim Kingdon, 23-Feb-2022.) |
Ref | Expression |
---|---|
fihasheq0 | ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0fin 6529 | . . 3 ⊢ ∅ ∈ Fin | |
2 | hashen 10026 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) | |
3 | 1, 2 | mpan2 416 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) |
4 | fz10 9354 | . . . . 5 ⊢ (1...0) = ∅ | |
5 | 4 | fveq2i 5255 | . . . 4 ⊢ (♯‘(1...0)) = (♯‘∅) |
6 | 0nn0 8579 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
7 | hashfz1 10025 | . . . . 5 ⊢ (0 ∈ ℕ0 → (♯‘(1...0)) = 0) | |
8 | 6, 7 | ax-mp 7 | . . . 4 ⊢ (♯‘(1...0)) = 0 |
9 | 5, 8 | eqtr3i 2105 | . . 3 ⊢ (♯‘∅) = 0 |
10 | 9 | eqeq2i 2093 | . 2 ⊢ ((♯‘𝐴) = (♯‘∅) ↔ (♯‘𝐴) = 0) |
11 | en0 6441 | . 2 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
12 | 3, 10, 11 | 3bitr3g 220 | 1 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1285 ∈ wcel 1434 ∅c0 3269 class class class wbr 3811 ‘cfv 4968 (class class class)co 5590 ≈ cen 6384 Fincfn 6386 0cc0 7252 1c1 7253 ℕ0cn0 8564 ...cfz 9318 ♯chash 10017 |
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 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 ax-cnex 7338 ax-resscn 7339 ax-1cn 7340 ax-1re 7341 ax-icn 7342 ax-addcl 7343 ax-addrcl 7344 ax-mulcl 7345 ax-addcom 7347 ax-addass 7349 ax-distr 7351 ax-i2m1 7352 ax-0lt1 7353 ax-0id 7355 ax-rnegex 7356 ax-cnre 7358 ax-pre-ltirr 7359 ax-pre-ltwlin 7360 ax-pre-lttrn 7361 ax-pre-apti 7362 ax-pre-ltadd 7363 |
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 4083 df-iord 4156 df-on 4158 df-ilim 4159 df-suc 4161 df-iom 4368 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-rn 4411 df-res 4412 df-ima 4413 df-iota 4933 df-fun 4970 df-fn 4971 df-f 4972 df-f1 4973 df-fo 4974 df-f1o 4975 df-fv 4976 df-riota 5546 df-ov 5593 df-oprab 5594 df-mpt2 5595 df-recs 6001 df-frec 6087 df-1o 6112 df-er 6221 df-en 6387 df-dom 6388 df-fin 6389 df-pnf 7426 df-mnf 7427 df-xr 7428 df-ltxr 7429 df-le 7430 df-sub 7557 df-neg 7558 df-inn 8316 df-n0 8565 df-z 8646 df-uz 8914 df-fz 9319 df-ihash 10018 |
This theorem is referenced by: fihashneq0 10037 hashnncl 10038 hash0 10039 fz1f1o 10571 |
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