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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 6778 | . . 3 ⊢ ∅ ∈ Fin | |
2 | hashen 10530 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) | |
3 | 1, 2 | mpan2 421 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) |
4 | fz10 9826 | . . . . 5 ⊢ (1...0) = ∅ | |
5 | 4 | fveq2i 5424 | . . . 4 ⊢ (♯‘(1...0)) = (♯‘∅) |
6 | 0nn0 8992 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
7 | hashfz1 10529 | . . . . 5 ⊢ (0 ∈ ℕ0 → (♯‘(1...0)) = 0) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ (♯‘(1...0)) = 0 |
9 | 5, 8 | eqtr3i 2162 | . . 3 ⊢ (♯‘∅) = 0 |
10 | 9 | eqeq2i 2150 | . 2 ⊢ ((♯‘𝐴) = (♯‘∅) ↔ (♯‘𝐴) = 0) |
11 | en0 6689 | . 2 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
12 | 3, 10, 11 | 3bitr3g 221 | 1 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∅c0 3363 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 ≈ cen 6632 Fincfn 6634 0cc0 7620 1c1 7621 ℕ0cn0 8977 ...cfz 9790 ♯chash 10521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-recs 6202 df-frec 6288 df-1o 6313 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 df-ihash 10522 |
This theorem is referenced by: fihashneq0 10541 hashnncl 10542 hash0 10543 fz1f1o 11144 |
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