hasheq0 - Metamath Proof Explorer

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Theorem hasheq0 14323

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 11255	. . . . . . 7 ⊢ +∞ ∉ ℝ
2	1	neli 3049	. . . . . 6 ⊢ ¬ +∞ ∈ ℝ
3		hashinf 14295	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
4	3	eleq1d 2819	. . . . . 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 11216	. . . . . 6 ⊢ 0 ∈ ℝ
8	6, 7	eqeltrdi 2842	. . . . 5 ⊢ ((♯‘𝐴) = 0 → (♯‘𝐴) ∈ ℝ)
9	5, 8	nsyl 140	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (♯‘𝐴) = 0)
10		id 22	. . . . . . 7 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
11		0fin 9171	. . . . . . 7 ⊢ ∅ ∈ Fin
12	10, 11	eqeltrdi 2842	. . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin)
13	12	con3i 154	. . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ¬ 𝐴 = ∅)
14	13	adantl 483	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 = ∅)
15	9, 14	2falsed 377	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
16	15	ex 414	. 2 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)))
17		hashen 14307	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅))
18	11, 17	mpan2 690	. . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅))
19		fz10 13522	. . . . . 6 ⊢ (1...0) = ∅
20	19	fveq2i 6895	. . . . 5 ⊢ (♯‘(1...0)) = (♯‘∅)
21		0nn0 12487	. . . . . 6 ⊢ 0 ∈ ℕ₀
22		hashfz1 14306	. . . . . 6 ⊢ (0 ∈ ℕ₀ → (♯‘(1...0)) = 0)
23	21, 22	ax-mp 5	. . . . 5 ⊢ (♯‘(1...0)) = 0
24	20, 23	eqtr3i 2763	. . . 4 ⊢ (♯‘∅) = 0
25	24	eqeq2i 2746	. . 3 ⊢ ((♯‘𝐴) = (♯‘∅) ↔ (♯‘𝐴) = 0)
26		en0 9013	. . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∅c0 4323 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ≈ cen 8936 Fincfn 8939 ℝcr 11109 0cc0 11110 1c1 11111 +∞cpnf 11245 ℕ₀cn0 12472 ...cfz 13484 ♯chash 14290

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-hash 14291

This theorem is referenced by: hashneq0 14324 hashnncl 14326 hash0 14327 hashelne0d 14328 hashgt0 14348 hashle00 14360 seqcoll2 14426 prprrab 14434 hashle2pr 14438 hashge2el2difr 14442 ccat0 14526 ccat1st1st 14578 wrdind 14672 wrd2ind 14673 swrdccat3blem 14689 rev0 14714 repsw0 14727 cshwidx0 14756 fz1f1o 15656 hashbc0 16938 0hashbc 16940 ram0 16955 cshws0 17035 symgvalstruct 19264 symgvalstructOLD 19265 gsmsymgrfix 19296 sylow1lem1 19466 sylow1lem4 19469 sylow2blem3 19490 frgpnabllem1 19741 0ringnnzr 20302 01eq0ringOLD 20306 vieta1lem2 25824 tgldimor 27784 uhgr0vsize0 28527 uhgr0edgfi 28528 usgr1v0e 28614 fusgrfisbase 28616 vtxd0nedgb 28776 vtxdusgr0edgnelALT 28784 usgrvd0nedg 28821 vtxdginducedm1lem4 28830 finsumvtxdg2size 28838 cyclnspth 29088 iswwlksnx 29125 umgrclwwlkge2 29275 clwwisshclwws 29299 hashecclwwlkn1 29361 umgrhashecclwwlk 29362 vdn0conngrumgrv2 29480 frgrwopreg 29607 frrusgrord0lem 29623 wlkl0 29651 frgrregord013 29679 frgrregord13 29680 frgrogt3nreg 29681 friendshipgt3 29682 wrdt2ind 32148 tocyc01 32308 lvecdim0i 32721 hasheuni 33114 signstfvn 33611 signstfveq0a 33618 signshnz 33633 spthcycl 34151 usgrgt2cycl 34152 acycgr1v 34171 umgracycusgr 34176 cusgracyclt3v 34178 elmrsubrn 34542 fsuppind 41210 lindsrng01 47197

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