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Theorem hashnncl 10962

Description: Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)

**Assertion**
Ref	Expression
hashnncl	⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅))

**Proof of Theorem hashnncl**
Step	Hyp	Ref	Expression
1		simpr 110	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ∈ ℕ)
2		nnne0 9084	. . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≠ 0)
3	2	adantl 277	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ≠ 0)
4		fihasheq0 10960	. . . . . 6 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
5	4	necon3bid 2418	. . . . 5 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ≠ 0 ↔ 𝐴 ≠ ∅))
6	5	adantr 276	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (♯‘𝐴) ∈ ℕ) → ((♯‘𝐴) ≠ 0 ↔ 𝐴 ≠ ∅))
7	3, 6	mpbid 147	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (♯‘𝐴) ∈ ℕ) → 𝐴 ≠ ∅)
8	1, 7	2thd 175	. 2 ⊢ ((𝐴 ∈ Fin ∧ (♯‘𝐴) ∈ ℕ) → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅))
9	2	necon2bi 2432	. . . 4 ⊢ ((♯‘𝐴) = 0 → ¬ (♯‘𝐴) ∈ ℕ)
10	9	adantl 277	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (♯‘𝐴) = 0) → ¬ (♯‘𝐴) ∈ ℕ)
11	4	biimpa 296	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (♯‘𝐴) = 0) → 𝐴 = ∅)
12		nner 2381	. . . 4 ⊢ (𝐴 = ∅ → ¬ 𝐴 ≠ ∅)
13	11, 12	syl 14	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (♯‘𝐴) = 0) → ¬ 𝐴 ≠ ∅)
14	10, 13	2falsed 704	. 2 ⊢ ((𝐴 ∈ Fin ∧ (♯‘𝐴) = 0) → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅))
15		hashcl 10948	. . 3 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
16		elnn0 9317	. . 3 ⊢ ((♯‘𝐴) ∈ ℕ₀ ↔ ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0))
17	15, 16	sylib 122	. 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0))
18	8, 14, 17	mpjaodan 800	1 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 710 = wceq 1373 ∈ wcel 2177 ≠ wne 2377 ∅c0 3464 ‘cfv 5280 Fincfn 6840 0cc0 7945 ℕcn 9056 ℕ₀cn0 9315 ♯chash 10942

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4167 ax-sep 4170 ax-nul 4178 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-setind 4593 ax-iinf 4644 ax-cnex 8036 ax-resscn 8037 ax-1cn 8038 ax-1re 8039 ax-icn 8040 ax-addcl 8041 ax-addrcl 8042 ax-mulcl 8043 ax-addcom 8045 ax-addass 8047 ax-distr 8049 ax-i2m1 8050 ax-0lt1 8051 ax-0id 8053 ax-rnegex 8054 ax-cnre 8056 ax-pre-ltirr 8057 ax-pre-ltwlin 8058 ax-pre-lttrn 8059 ax-pre-apti 8060 ax-pre-ltadd 8061

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-nul 3465 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-int 3892 df-iun 3935 df-br 4052 df-opab 4114 df-mpt 4115 df-tr 4151 df-id 4348 df-iord 4421 df-on 4423 df-ilim 4424 df-suc 4426 df-iom 4647 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-rn 4694 df-res 4695 df-ima 4696 df-iota 5241 df-fun 5282 df-fn 5283 df-f 5284 df-f1 5285 df-fo 5286 df-f1o 5287 df-fv 5288 df-riota 5912 df-ov 5960 df-oprab 5961 df-mpo 5962 df-recs 6404 df-frec 6490 df-1o 6515 df-er 6633 df-en 6841 df-dom 6842 df-fin 6843 df-pnf 8129 df-mnf 8130 df-xr 8131 df-ltxr 8132 df-le 8133 df-sub 8265 df-neg 8266 df-inn 9057 df-n0 9316 df-z 9393 df-uz 9669 df-fz 10151 df-ihash 10943

This theorem is referenced by: 1elfz0hash 10973 lennncl 11036 lswlgt0cl 11068 wrdind 11198 wrd2ind 11199 gsumwmhm 13405

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