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 3040	. . . . . 6 ⊢ ¬ +∞ ∈ ℝ
3		hashinf 14288	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
4	3	eleq1d 2824	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((♯‘𝐴) ∈ ℝ ↔ +∞ ∈ ℝ))
5	2, 4	mtbiri 328	. . . . 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 2847	. . . . 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 8979	. . . . . . 7 ⊢ ∅ ∈ Fin
12	10, 11	eqeltrdi 2847	. . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin)
13	12	con3i 154	. . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ¬ 𝐴 = ∅)
14	13	adantl 482	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 = ∅)
15	9, 14	2falsed 377	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
16	15	ex 413	. 2 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)))
17		hashen 14300	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅))
18	11, 17	mpan2 697	. . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅))
19		fz10 13490	. . . . . 6 ⊢ (1...0) = ∅
20	19	fveq2i 6830	. . . . 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 2764	. . . 4 ⊢ (♯‘∅) = 0
25	24	eqeq2i 2752	. . 3 ⊢ ((♯‘𝐴) = (♯‘∅) ↔ (♯‘𝐴) = 0)
26		en0 8955	. . 3 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
27	18, 25, 26	3bitr3g 314	. 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
28	16, 27	pm2.61d2 182	1 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∅c0 4261 class class class wbr 5072 ‘cfv 6485 (class class class)co 7356 ≈ cen 8880 Fincfn 8883 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 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 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 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 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 20497 01eq0ringOLD 20503 vieta1lem2 26295 tgldimor 28588 uhgr0vsize0 29326 uhgr0edgfi 29327 usgr1v0e 29413 fusgrfisbase 29415 vtxd0nedgb 29575 vtxdusgr0edgnelALT 29583 usgrvd0nedg 29620 vtxdginducedm1lem4 29629 finsumvtxdg2size 29637 cyclnspth 29887 iswwlksnx 29926 umgrclwwlkge2 30079 clwwisshclwws 30103 hashecclwwlkn1 30165 umgrhashecclwwlk 30166 vdn0conngrumgrv2 30284 frgrwopreg 30411 frrusgrord0lem 30427 wlkl0 30455 frgrregord013 30483 frgrregord13 30484 frgrogt3nreg 30485 friendshipgt3 30486 hashne0 32902 wrdt2ind 33032 tocyc01 33199 esplyfval0 33748 vieta 33764 lvecdim0i 33790 hasheuni 34269 signstfvn 34753 signstfveq0a 34760 signshnz 34775 spthcycl 35357 usgrgt2cycl 35358 acycgr1v 35377 umgracycusgr 35382 cusgracyclt3v 35384 elmrsubrn 35748 fsuppind 43040 lindsrng01 48959

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