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Mirrors > Home > MPE Home > Th. List > hasheq0 | Structured version Visualization version GIF version |
Description: Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.) |
Ref | Expression |
---|---|
hasheq0 | ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfnre 10671 | . . . . . . 7 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 3125 | . . . . . 6 ⊢ ¬ +∞ ∈ ℝ |
3 | hashinf 13685 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | |
4 | 3 | eleq1d 2897 | . . . . . 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 10632 | . . . . . 6 ⊢ 0 ∈ ℝ | |
8 | 6, 7 | syl6eqel 2921 | . . . . 5 ⊢ ((♯‘𝐴) = 0 → (♯‘𝐴) ∈ ℝ) |
9 | 5, 8 | nsyl 142 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (♯‘𝐴) = 0) |
10 | id 22 | . . . . . . 7 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
11 | 0fin 8735 | . . . . . . 7 ⊢ ∅ ∈ Fin | |
12 | 10, 11 | syl6eqel 2921 | . . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin) |
13 | 12 | con3i 157 | . . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ¬ 𝐴 = ∅) |
14 | 13 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 = ∅) |
15 | 9, 14 | 2falsed 378 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) |
16 | 15 | ex 413 | . 2 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))) |
17 | hashen 13697 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) | |
18 | 11, 17 | mpan2 687 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) |
19 | fz10 12918 | . . . . . 6 ⊢ (1...0) = ∅ | |
20 | 19 | fveq2i 6667 | . . . . 5 ⊢ (♯‘(1...0)) = (♯‘∅) |
21 | 0nn0 11901 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
22 | hashfz1 13696 | . . . . . 6 ⊢ (0 ∈ ℕ0 → (♯‘(1...0)) = 0) | |
23 | 21, 22 | ax-mp 5 | . . . . 5 ⊢ (♯‘(1...0)) = 0 |
24 | 20, 23 | eqtr3i 2846 | . . . 4 ⊢ (♯‘∅) = 0 |
25 | 24 | eqeq2i 2834 | . . 3 ⊢ ((♯‘𝐴) = (♯‘∅) ↔ (♯‘𝐴) = 0) |
26 | en0 8561 | . . 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 1528 ∈ wcel 2105 ∅c0 4290 class class class wbr 5058 ‘cfv 6349 (class class class)co 7145 ≈ cen 8495 Fincfn 8498 ℝcr 10525 0cc0 10526 1c1 10527 +∞cpnf 10661 ℕ0cn0 11886 ...cfz 12882 ♯chash 13680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-n0 11887 df-z 11971 df-uz 12233 df-fz 12883 df-hash 13681 |
This theorem is referenced by: hashneq0 13715 hashnncl 13717 hash0 13718 hashelne0d 13719 hashgt0 13739 hashle00 13751 seqcoll2 13813 prprrab 13821 hashle2pr 13825 hashge2el2difr 13829 ccat0 13919 ccat1st1st 13974 wrdind 14074 wrd2ind 14075 swrdccat3blem 14091 rev0 14116 repsw0 14129 cshwidx0 14158 fz1f1o 15057 hashbc0 16331 0hashbc 16333 ram0 16348 cshws0 16425 gsmsymgrfix 18487 sylow1lem1 18654 sylow1lem4 18657 sylow2blem3 18678 frgpnabllem1 18924 0ringnnzr 19972 01eq0ring 19975 vieta1lem2 24829 tgldimor 26216 uhgr0vsize0 26949 uhgr0edgfi 26950 usgr1v0e 27036 fusgrfisbase 27038 vtxd0nedgb 27198 vtxdusgr0edgnelALT 27206 usgrvd0nedg 27243 vtxdginducedm1lem4 27252 finsumvtxdg2size 27260 cyclnspth 27509 iswwlksnx 27546 umgrclwwlkge2 27697 clwwisshclwws 27721 hashecclwwlkn1 27784 umgrhashecclwwlk 27785 vdn0conngrumgrv2 27903 frgrwopreg 28030 frrusgrord0lem 28046 wlkl0 28074 frgrregord013 28102 frgrregord13 28103 frgrogt3nreg 28104 friendshipgt3 28105 wrdt2ind 30555 tocyc01 30688 lvecdim0i 30904 hasheuni 31244 signstfvn 31739 signstfveq0a 31746 signshnz 31761 spthcycl 32274 usgrgt2cycl 32275 acycgr1v 32294 umgracycusgr 32299 cusgracyclt3v 32301 elmrsubrn 32665 lindsrng01 44421 |
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