hasheq0 - Metamath Proof Explorer

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Theorem hasheq0 14286

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 11173	. . . . . . 7 ⊢ +∞ ∉ ℝ
2	1	neli 3038	. . . . . 6 ⊢ ¬ +∞ ∈ ℝ
3		hashinf 14258	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
4	3	eleq1d 2821	. . . . . 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 11134	. . . . . 6 ⊢ 0 ∈ ℝ
8	6, 7	eqeltrdi 2844	. . . . 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 2844	. . . . . 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 14270	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅))
18	11, 17	mpan2 691	. . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅))
19		fz10 13461	. . . . . 6 ⊢ (1...0) = ∅
20	19	fveq2i 6837	. . . . 5 ⊢ (♯‘(1...0)) = (♯‘∅)
21		0nn0 12416	. . . . . 6 ⊢ 0 ∈ ℕ₀
22		hashfz1 14269	. . . . . 6 ⊢ (0 ∈ ℕ₀ → (♯‘(1...0)) = 0)
23	21, 22	ax-mp 5	. . . . 5 ⊢ (♯‘(1...0)) = 0
24	20, 23	eqtr3i 2761	. . . 4 ⊢ (♯‘∅) = 0
25	24	eqeq2i 2749	. . 3 ⊢ ((♯‘𝐴) = (♯‘∅) ↔ (♯‘𝐴) = 0)
26		en0 8955	. . 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 1541 ∈ wcel 2113 ∅c0 4285 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 ≈ cen 8880 Fincfn 8883 ℝcr 11025 0cc0 11026 1c1 11027 +∞cpnf 11163 ℕ₀cn0 12401 ...cfz 13423 ♯chash 14253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 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 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-n0 12402 df-z 12489 df-uz 12752 df-fz 13424 df-hash 14254

This theorem is referenced by: hashneq0 14287 hashnncl 14289 hash0 14290 hashelne0d 14291 hashgt0 14311 hashle00 14323 seqcoll2 14388 prprrab 14396 hashle2pr 14400 hashge2el2difr 14404 ccat0 14499 ccat1st1st 14552 wrdind 14645 wrd2ind 14646 swrdccat3blem 14662 rev0 14687 repsw0 14700 cshwidx0 14729 fz1f1o 15633 hashbc0 16933 0hashbc 16935 ram0 16950 cshws0 17029 chnind 18544 chnub 18545 symgvalstruct 19326 gsmsymgrfix 19357 sylow1lem1 19527 sylow1lem4 19530 sylow2blem3 19551 frgpnabllem1 19802 0ringnnzr 20458 01eq0ringOLD 20464 vieta1lem2 26275 tgldimor 28574 uhgr0vsize0 29312 uhgr0edgfi 29313 usgr1v0e 29399 fusgrfisbase 29401 vtxd0nedgb 29562 vtxdusgr0edgnelALT 29570 usgrvd0nedg 29607 vtxdginducedm1lem4 29616 finsumvtxdg2size 29624 cyclnspth 29874 iswwlksnx 29913 umgrclwwlkge2 30066 clwwisshclwws 30090 hashecclwwlkn1 30152 umgrhashecclwwlk 30153 vdn0conngrumgrv2 30271 frgrwopreg 30398 frrusgrord0lem 30414 wlkl0 30442 frgrregord013 30470 frgrregord13 30471 frgrogt3nreg 30472 friendshipgt3 30473 hashne0 32890 wrdt2ind 33035 tocyc01 33200 esplyfval0 33722 vieta 33736 lvecdim0i 33762 hasheuni 34242 signstfvn 34726 signstfveq0a 34733 signshnz 34748 spthcycl 35323 usgrgt2cycl 35324 acycgr1v 35343 umgracycusgr 35348 cusgracyclt3v 35350 elmrsubrn 35714 fsuppind 42833 lindsrng01 48714

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