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Mirrors > Home > ILE Home > Th. List > fihashneq0 | GIF version |
Description: Two ways of saying a finite set is not empty. Also, "A is inhabited" would be equivalent by fin0 6914. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Intuitionized by Jim Kingdon, 23-Feb-2022.) |
Ref | Expression |
---|---|
fihashneq0 | ⊢ (𝐴 ∈ Fin → (0 < (♯‘𝐴) ↔ 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashcl 10796 | . . . 4 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
2 | 1 | nn0zd 9404 | . . 3 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℤ) |
3 | 0zd 9296 | . . 3 ⊢ (𝐴 ∈ Fin → 0 ∈ ℤ) | |
4 | zapne 9358 | . . 3 ⊢ (((♯‘𝐴) ∈ ℤ ∧ 0 ∈ ℤ) → ((♯‘𝐴) # 0 ↔ (♯‘𝐴) ≠ 0)) | |
5 | 2, 3, 4 | syl2anc 411 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) # 0 ↔ (♯‘𝐴) ≠ 0)) |
6 | nn0re 9216 | . . . 4 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℝ) | |
7 | nn0ge0 9232 | . . . 4 ⊢ ((♯‘𝐴) ∈ ℕ0 → 0 ≤ (♯‘𝐴)) | |
8 | ap0gt0 8628 | . . . 4 ⊢ (((♯‘𝐴) ∈ ℝ ∧ 0 ≤ (♯‘𝐴)) → ((♯‘𝐴) # 0 ↔ 0 < (♯‘𝐴))) | |
9 | 6, 7, 8 | syl2anc 411 | . . 3 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴) # 0 ↔ 0 < (♯‘𝐴))) |
10 | 1, 9 | syl 14 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) # 0 ↔ 0 < (♯‘𝐴))) |
11 | fihasheq0 10808 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) | |
12 | 11 | necon3bid 2401 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ≠ 0 ↔ 𝐴 ≠ ∅)) |
13 | 5, 10, 12 | 3bitr3d 218 | 1 ⊢ (𝐴 ∈ Fin → (0 < (♯‘𝐴) ↔ 𝐴 ≠ ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∈ wcel 2160 ≠ wne 2360 ∅c0 3437 class class class wbr 4018 ‘cfv 5235 Fincfn 6767 ℝcr 7841 0cc0 7842 < clt 8023 ≤ cle 8024 # cap 8569 ℕ0cn0 9207 ℤcz 9284 ♯chash 10790 |
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 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 |
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 4311 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-recs 6331 df-frec 6417 df-1o 6442 df-er 6560 df-en 6768 df-dom 6769 df-fin 6770 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-inn 8951 df-n0 9208 df-z 9285 df-uz 9560 df-fz 10041 df-ihash 10791 |
This theorem is referenced by: (None) |
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