hasheq0 - Metamath Proof Explorer

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Theorem hasheq0 14316

Description: Two ways of saying a 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 11177	. . . . . . 7 ⊢ +∞ ∉ ℝ
2	1	neli 3039	. . . . . 6 ⊢ ¬ +∞ ∈ ℝ
3		hashinf 14288	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
4	3	eleq1d 2822	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((♯‘𝐴) ∈ ℝ ↔ +∞ ∈ ℝ))
5	2, 4	mtbiri 327	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (♯‘𝐴) ∈ ℝ)
6		id 22	. . . . . 6 ⊢ ((♯‘𝐴) = 0 → (♯‘𝐴) = 0)
7		0re 11137	. . . . . 6 ⊢ 0 ∈ ℝ
8	6, 7	eqeltrdi 2845	. . . . 5 ⊢ ((♯‘𝐴) = 0 → (♯‘𝐴) ∈ ℝ)
9	5, 8	nsyl 140	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (♯‘𝐴) = 0)
10		id 22	. . . . . . 7 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
11		0fi 8982	. . . . . . 7 ⊢ ∅ ∈ Fin
12	10, 11	eqeltrdi 2845	. . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin)
13	12	con3i 154	. . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ¬ 𝐴 = ∅)
14	13	adantl 481	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 = ∅)
15	9, 14	2falsed 376	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
16	15	ex 412	. 2 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)))
17		hashen 14300	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅))
18	11, 17	mpan2 692	. . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅))
19		fz10 13490	. . . . . 6 ⊢ (1...0) = ∅
20	19	fveq2i 6837	. . . . 5 ⊢ (♯‘(1...0)) = (♯‘∅)
21		0nn0 12443	. . . . . 6 ⊢ 0 ∈ ℕ₀
22		hashfz1 14299	. . . . . 6 ⊢ (0 ∈ ℕ₀ → (♯‘(1...0)) = 0)
23	21, 22	ax-mp 5	. . . . 5 ⊢ (♯‘(1...0)) = 0
24	20, 23	eqtr3i 2762	. . . 4 ⊢ (♯‘∅) = 0
25	24	eqeq2i 2750	. . 3 ⊢ ((♯‘𝐴) = (♯‘∅) ↔ (♯‘𝐴) = 0)
26		en0 8958	. . 3 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
27	18, 25, 26	3bitr3g 313	. 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
28	16, 27	pm2.61d2 181	1 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∅c0 4274 class class class wbr 5086 ‘cfv 6492 (class class class)co 7360 ≈ cen 8883 Fincfn 8886 ℝcr 11028 0cc0 11029 1c1 11030 +∞cpnf 11167 ℕ₀cn0 12428 ...cfz 13452 ♯chash 14283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-hash 14284

This theorem is referenced by: hashneq0 14317 hashnncl 14319 hash0 14320 hashelne0d 14321 hashgt0 14341 hashle00 14353 seqcoll2 14418 prprrab 14426 hashle2pr 14430 hashge2el2difr 14434 ccat0 14529 ccat1st1st 14582 wrdind 14675 wrd2ind 14676 swrdccat3blem 14692 rev0 14717 repsw0 14730 cshwidx0 14759 fz1f1o 15663 hashbc0 16967 0hashbc 16969 ram0 16984 cshws0 17063 chnind 18578 chnub 18579 symgvalstruct 19363 gsmsymgrfix 19394 sylow1lem1 19564 sylow1lem4 19567 sylow2blem3 19588 frgpnabllem1 19839 0ringnnzr 20493 01eq0ringOLD 20499 vieta1lem2 26288 tgldimor 28584 uhgr0vsize0 29322 uhgr0edgfi 29323 usgr1v0e 29409 fusgrfisbase 29411 vtxd0nedgb 29572 vtxdusgr0edgnelALT 29580 usgrvd0nedg 29617 vtxdginducedm1lem4 29626 finsumvtxdg2size 29634 cyclnspth 29884 iswwlksnx 29923 umgrclwwlkge2 30076 clwwisshclwws 30100 hashecclwwlkn1 30162 umgrhashecclwwlk 30163 vdn0conngrumgrv2 30281 frgrwopreg 30408 frrusgrord0lem 30424 wlkl0 30452 frgrregord013 30480 frgrregord13 30481 frgrogt3nreg 30482 friendshipgt3 30483 hashne0 32898 wrdt2ind 33028 tocyc01 33194 esplyfval0 33723 vieta 33739 lvecdim0i 33765 hasheuni 34245 signstfvn 34729 signstfveq0a 34736 signshnz 34751 spthcycl 35327 usgrgt2cycl 35328 acycgr1v 35347 umgracycusgr 35352 cusgracyclt3v 35354 elmrsubrn 35718 fsuppind 43037 lindsrng01 48956

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