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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 11016 | . . . . . . 7 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 3051 | . . . . . 6 ⊢ ¬ +∞ ∈ ℝ |
3 | hashinf 14049 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | |
4 | 3 | eleq1d 2823 | . . . . . 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 10977 | . . . . . 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 | 0fin 8954 | . . . . . . 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 14061 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) | |
18 | 11, 17 | mpan2 688 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) |
19 | fz10 13277 | . . . . . 6 ⊢ (1...0) = ∅ | |
20 | 19 | fveq2i 6777 | . . . . 5 ⊢ (♯‘(1...0)) = (♯‘∅) |
21 | 0nn0 12248 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
22 | hashfz1 14060 | . . . . . 6 ⊢ (0 ∈ ℕ0 → (♯‘(1...0)) = 0) | |
23 | 21, 22 | ax-mp 5 | . . . . 5 ⊢ (♯‘(1...0)) = 0 |
24 | 20, 23 | eqtr3i 2768 | . . . 4 ⊢ (♯‘∅) = 0 |
25 | 24 | eqeq2i 2751 | . . 3 ⊢ ((♯‘𝐴) = (♯‘∅) ↔ (♯‘𝐴) = 0) |
26 | en0 8803 | . . 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 396 = wceq 1539 ∈ wcel 2106 ∅c0 4256 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ≈ cen 8730 Fincfn 8733 ℝcr 10870 0cc0 10871 1c1 10872 +∞cpnf 11006 ℕ0cn0 12233 ...cfz 13239 ♯chash 14044 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-hash 14045 |
This theorem is referenced by: hashneq0 14079 hashnncl 14081 hash0 14082 hashelne0d 14083 hashgt0 14103 hashle00 14115 seqcoll2 14179 prprrab 14187 hashle2pr 14191 hashge2el2difr 14195 ccat0 14280 ccat1st1st 14335 wrdind 14435 wrd2ind 14436 swrdccat3blem 14452 rev0 14477 repsw0 14490 cshwidx0 14519 fz1f1o 15422 hashbc0 16706 0hashbc 16708 ram0 16723 cshws0 16803 symgvalstruct 19004 symgvalstructOLD 19005 gsmsymgrfix 19036 sylow1lem1 19203 sylow1lem4 19206 sylow2blem3 19227 frgpnabllem1 19474 0ringnnzr 20540 01eq0ring 20543 vieta1lem2 25471 tgldimor 26863 uhgr0vsize0 27606 uhgr0edgfi 27607 usgr1v0e 27693 fusgrfisbase 27695 vtxd0nedgb 27855 vtxdusgr0edgnelALT 27863 usgrvd0nedg 27900 vtxdginducedm1lem4 27909 finsumvtxdg2size 27917 cyclnspth 28168 iswwlksnx 28205 umgrclwwlkge2 28355 clwwisshclwws 28379 hashecclwwlkn1 28441 umgrhashecclwwlk 28442 vdn0conngrumgrv2 28560 frgrwopreg 28687 frrusgrord0lem 28703 wlkl0 28731 frgrregord013 28759 frgrregord13 28760 frgrogt3nreg 28761 friendshipgt3 28762 wrdt2ind 31225 tocyc01 31385 lvecdim0i 31689 hasheuni 32053 signstfvn 32548 signstfveq0a 32555 signshnz 32570 spthcycl 33091 usgrgt2cycl 33092 acycgr1v 33111 umgracycusgr 33116 cusgracyclt3v 33118 elmrsubrn 33482 fsuppind 40279 lindsrng01 45809 |
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