hasheq0 - Metamath Proof Explorer

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Theorem hasheq0 13473

Description: Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.)

**Assertion**
Ref	Expression
hasheq0	⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))

**Proof of Theorem hasheq0**
Step	Hyp	Ref	Expression
1		pnfnre 10420	. . . . . . 7 ⊢ +∞ ∉ ℝ
2	1	neli 3077	. . . . . 6 ⊢ ¬ +∞ ∈ ℝ
3		hashinf 13444	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
4	3	eleq1d 2844	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((♯‘𝐴) ∈ ℝ ↔ +∞ ∈ ℝ))
5	2, 4	mtbiri 319	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (♯‘𝐴) ∈ ℝ)
6		id 22	. . . . . 6 ⊢ ((♯‘𝐴) = 0 → (♯‘𝐴) = 0)
7		0re 10380	. . . . . 6 ⊢ 0 ∈ ℝ
8	6, 7	syl6eqel 2867	. . . . 5 ⊢ ((♯‘𝐴) = 0 → (♯‘𝐴) ∈ ℝ)
9	5, 8	nsyl 138	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (♯‘𝐴) = 0)
10		id 22	. . . . . . 7 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
11		0fin 8478	. . . . . . 7 ⊢ ∅ ∈ Fin
12	10, 11	syl6eqel 2867	. . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin)
13	12	con3i 152	. . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ¬ 𝐴 = ∅)
14	13	adantl 475	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 = ∅)
15	9, 14	2falsed 368	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
16	15	ex 403	. 2 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)))
17		hashen 13456	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅))
18	11, 17	mpan2 681	. . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅))
19		fz10 12683	. . . . . 6 ⊢ (1...0) = ∅
20	19	fveq2i 6451	. . . . 5 ⊢ (♯‘(1...0)) = (♯‘∅)
21		0nn0 11663	. . . . . 6 ⊢ 0 ∈ ℕ₀
22		hashfz1 13455	. . . . . 6 ⊢ (0 ∈ ℕ₀ → (♯‘(1...0)) = 0)
23	21, 22	ax-mp 5	. . . . 5 ⊢ (♯‘(1...0)) = 0
24	20, 23	eqtr3i 2804	. . . 4 ⊢ (♯‘∅) = 0
25	24	eqeq2i 2790	. . 3 ⊢ ((♯‘𝐴) = (♯‘∅) ↔ (♯‘𝐴) = 0)
26		en0 8306	. . 3 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
27	18, 25, 26	3bitr3g 305	. 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
28	16, 27	pm2.61d2 174	1 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∅c0 4141 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 ≈ cen 8240 Fincfn 8243 ℝcr 10273 0cc0 10274 1c1 10275 +∞cpnf 10410 ℕ₀cn0 11646 ...cfz 12647 ♯chash 13439

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-n0 11647 df-z 11733 df-uz 11997 df-fz 12648 df-hash 13440

This theorem is referenced by: hashneq0 13474 hashnncl 13476 hash0 13477 hashgt0 13496 hashle00 13506 seqcoll2 13567 prprrab 13573 hashle2pr 13577 hashge2el2difr 13581 ccat0 13670 ccat1st1st 13722 wrdind 13846 wrdindOLD 13847 wrd2ind 13848 wrd2indOLD 13849 swrdccat3aOLD 13874 swrdccat3blem 13875 rev0 13914 repsw0 13927 cshwidx0 13961 fz1f1o 14852 hashbc0 16117 0hashbc 16119 ram0 16134 cshws0 16211 gsmsymgrfix 18235 sylow1lem1 18401 sylow1lem4 18404 sylow2blem3 18425 frgpnabllem1 18666 0ringnnzr 19670 01eq0ring 19673 vieta1lem2 24507 tgldimor 25857 uhgr0vsize0 26590 uhgr0edgfi 26591 usgr1v0e 26677 fusgrfisbase 26679 vtxd0nedgb 26840 vtxdusgr0edgnelALT 26848 usgrvd0nedg 26885 vtxdginducedm1lem4 26894 finsumvtxdg2size 26902 cyclnspth 27156 iswwlksnx 27193 umgrclwwlkge2 27375 clwwisshclwws 27408 hashecclwwlkn1 27479 umgrhashecclwwlk 27480 vdn0conngrumgrv2 27603 frgrwopreg 27735 frrusgrord0lem 27751 wlkl0 27799 frgrregord013 27831 frgrregord13 27832 frgrogt3nreg 27833 friendshipgt3 27834 hasheuni 30749 signstfvn 31250 signstfveq0a 31258 signshnz 31274 elmrsubrn 32020 lindsrng01 43282

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