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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 6902 | . . 3 ⊢ ∅ ∈ Fin | |
2 | hashen 10783 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) | |
3 | 1, 2 | mpan2 425 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) |
4 | fz10 10065 | . . . . 5 ⊢ (1...0) = ∅ | |
5 | 4 | fveq2i 5533 | . . . 4 ⊢ (♯‘(1...0)) = (♯‘∅) |
6 | 0nn0 9210 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
7 | hashfz1 10782 | . . . . 5 ⊢ (0 ∈ ℕ0 → (♯‘(1...0)) = 0) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ (♯‘(1...0)) = 0 |
9 | 5, 8 | eqtr3i 2212 | . . 3 ⊢ (♯‘∅) = 0 |
10 | 9 | eqeq2i 2200 | . 2 ⊢ ((♯‘𝐴) = (♯‘∅) ↔ (♯‘𝐴) = 0) |
11 | en0 6813 | . 2 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
12 | 3, 10, 11 | 3bitr3g 222 | 1 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 ∅c0 3437 class class class wbr 4018 ‘cfv 5231 (class class class)co 5891 ≈ cen 6756 Fincfn 6758 0cc0 7830 1c1 7831 ℕ0cn0 9195 ...cfz 10027 ♯chash 10774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-addcom 7930 ax-addass 7932 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-0id 7938 ax-rnegex 7939 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-recs 6324 df-frec 6410 df-1o 6435 df-er 6553 df-en 6759 df-dom 6760 df-fin 6761 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-inn 8939 df-n0 9196 df-z 9273 df-uz 9548 df-fz 10028 df-ihash 10775 |
This theorem is referenced by: fihashneq0 10793 hashnncl 10794 hash0 10795 fihashelne0d 10796 fz1f1o 11402 |
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