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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 6841 | . . 3 ⊢ ∅ ∈ Fin | |
2 | hashen 10686 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) | |
3 | 1, 2 | mpan2 422 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) |
4 | fz10 9971 | . . . . 5 ⊢ (1...0) = ∅ | |
5 | 4 | fveq2i 5483 | . . . 4 ⊢ (♯‘(1...0)) = (♯‘∅) |
6 | 0nn0 9120 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
7 | hashfz1 10685 | . . . . 5 ⊢ (0 ∈ ℕ0 → (♯‘(1...0)) = 0) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ (♯‘(1...0)) = 0 |
9 | 5, 8 | eqtr3i 2187 | . . 3 ⊢ (♯‘∅) = 0 |
10 | 9 | eqeq2i 2175 | . 2 ⊢ ((♯‘𝐴) = (♯‘∅) ↔ (♯‘𝐴) = 0) |
11 | en0 6752 | . 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 1342 ∈ wcel 2135 ∅c0 3404 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ≈ cen 6695 Fincfn 6697 0cc0 7744 1c1 7745 ℕ0cn0 9105 ...cfz 9935 ♯chash 10677 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-recs 6264 df-frec 6350 df-1o 6375 df-er 6492 df-en 6698 df-dom 6699 df-fin 6700 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 df-ihash 10678 |
This theorem is referenced by: fihashneq0 10697 hashnncl 10698 hash0 10699 fz1f1o 11302 |
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