hasheq0 - Metamath Proof Explorer

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Theorem hasheq0 14304

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 11191	. . . . . . 7 ⊢ +∞ ∉ ℝ
2	1	neli 3031	. . . . . 6 ⊢ ¬ +∞ ∈ ℝ
3		hashinf 14276	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
4	3	eleq1d 2813	. . . . . 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 11152	. . . . . 6 ⊢ 0 ∈ ℝ
8	6, 7	eqeltrdi 2836	. . . . 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 8990	. . . . . . 7 ⊢ ∅ ∈ Fin
12	10, 11	eqeltrdi 2836	. . . . . 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 14288	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅))
18	11, 17	mpan2 691	. . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅))
19		fz10 13482	. . . . . 6 ⊢ (1...0) = ∅
20	19	fveq2i 6843	. . . . 5 ⊢ (♯‘(1...0)) = (♯‘∅)
21		0nn0 12433	. . . . . 6 ⊢ 0 ∈ ℕ₀
22		hashfz1 14287	. . . . . 6 ⊢ (0 ∈ ℕ₀ → (♯‘(1...0)) = 0)
23	21, 22	ax-mp 5	. . . . 5 ⊢ (♯‘(1...0)) = 0
24	20, 23	eqtr3i 2754	. . . 4 ⊢ (♯‘∅) = 0
25	24	eqeq2i 2742	. . 3 ⊢ ((♯‘𝐴) = (♯‘∅) ↔ (♯‘𝐴) = 0)
26		en0 8966	. . 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 1540 ∈ wcel 2109 ∅c0 4292 class class class wbr 5102 ‘cfv 6499 (class class class)co 7369 ≈ cen 8892 Fincfn 8895 ℝcr 11043 0cc0 11044 1c1 11045 +∞cpnf 11181 ℕ₀cn0 12418 ...cfz 13444 ♯chash 14271

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445 df-hash 14272

This theorem is referenced by: hashneq0 14305 hashnncl 14307 hash0 14308 hashelne0d 14309 hashgt0 14329 hashle00 14341 seqcoll2 14406 prprrab 14414 hashle2pr 14418 hashge2el2difr 14422 ccat0 14517 ccat1st1st 14569 wrdind 14663 wrd2ind 14664 swrdccat3blem 14680 rev0 14705 repsw0 14718 cshwidx0 14747 fz1f1o 15652 hashbc0 16952 0hashbc 16954 ram0 16969 cshws0 17048 symgvalstruct 19303 gsmsymgrfix 19334 sylow1lem1 19504 sylow1lem4 19507 sylow2blem3 19528 frgpnabllem1 19779 0ringnnzr 20410 01eq0ringOLD 20416 vieta1lem2 26195 tgldimor 28405 uhgr0vsize0 29142 uhgr0edgfi 29143 usgr1v0e 29229 fusgrfisbase 29231 vtxd0nedgb 29392 vtxdusgr0edgnelALT 29400 usgrvd0nedg 29437 vtxdginducedm1lem4 29446 finsumvtxdg2size 29454 cyclnspth 29704 iswwlksnx 29743 umgrclwwlkge2 29893 clwwisshclwws 29917 hashecclwwlkn1 29979 umgrhashecclwwlk 29980 vdn0conngrumgrv2 30098 frgrwopreg 30225 frrusgrord0lem 30241 wlkl0 30269 frgrregord013 30297 frgrregord13 30298 frgrogt3nreg 30299 friendshipgt3 30300 hashne0 32708 wrdt2ind 32848 chnind 32910 chnub 32911 tocyc01 33048 lvecdim0i 33574 hasheuni 34048 signstfvn 34533 signstfveq0a 34540 signshnz 34555 spthcycl 35089 usgrgt2cycl 35090 acycgr1v 35109 umgracycusgr 35114 cusgracyclt3v 35116 elmrsubrn 35480 fsuppind 42551 lindsrng01 48430

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