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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 10119 | . . . . . . 7 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 2928 | . . . . . 6 ⊢ ¬ +∞ ∈ ℝ |
3 | hashinf 13162 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞) | |
4 | 3 | eleq1d 2715 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((#‘𝐴) ∈ ℝ ↔ +∞ ∈ ℝ)) |
5 | 2, 4 | mtbiri 316 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (#‘𝐴) ∈ ℝ) |
6 | id 22 | . . . . . 6 ⊢ ((#‘𝐴) = 0 → (#‘𝐴) = 0) | |
7 | 0re 10078 | . . . . . 6 ⊢ 0 ∈ ℝ | |
8 | 6, 7 | syl6eqel 2738 | . . . . 5 ⊢ ((#‘𝐴) = 0 → (#‘𝐴) ∈ ℝ) |
9 | 5, 8 | nsyl 135 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (#‘𝐴) = 0) |
10 | id 22 | . . . . . . 7 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
11 | 0fin 8229 | . . . . . . 7 ⊢ ∅ ∈ Fin | |
12 | 10, 11 | syl6eqel 2738 | . . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin) |
13 | 12 | con3i 150 | . . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ¬ 𝐴 = ∅) |
14 | 13 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 = ∅) |
15 | 9, 14 | 2falsed 365 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((#‘𝐴) = 0 ↔ 𝐴 = ∅)) |
16 | 15 | ex 449 | . 2 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ Fin → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))) |
17 | hashen 13175 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((#‘𝐴) = (#‘∅) ↔ 𝐴 ≈ ∅)) | |
18 | 11, 17 | mpan2 707 | . . 3 ⊢ (𝐴 ∈ Fin → ((#‘𝐴) = (#‘∅) ↔ 𝐴 ≈ ∅)) |
19 | fz10 12400 | . . . . . 6 ⊢ (1...0) = ∅ | |
20 | 19 | fveq2i 6232 | . . . . 5 ⊢ (#‘(1...0)) = (#‘∅) |
21 | 0nn0 11345 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
22 | hashfz1 13174 | . . . . . 6 ⊢ (0 ∈ ℕ0 → (#‘(1...0)) = 0) | |
23 | 21, 22 | ax-mp 5 | . . . . 5 ⊢ (#‘(1...0)) = 0 |
24 | 20, 23 | eqtr3i 2675 | . . . 4 ⊢ (#‘∅) = 0 |
25 | 24 | eqeq2i 2663 | . . 3 ⊢ ((#‘𝐴) = (#‘∅) ↔ (#‘𝐴) = 0) |
26 | en0 8060 | . . 3 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
27 | 18, 25, 26 | 3bitr3g 302 | . 2 ⊢ (𝐴 ∈ Fin → ((#‘𝐴) = 0 ↔ 𝐴 = ∅)) |
28 | 16, 27 | pm2.61d2 172 | 1 ⊢ (𝐴 ∈ 𝑉 → ((#‘𝐴) = 0 ↔ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∅c0 3948 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 ≈ cen 7994 Fincfn 7997 ℝcr 9973 0cc0 9974 1c1 9975 +∞cpnf 10109 ℕ0cn0 11330 ...cfz 12364 #chash 13157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-hash 13158 |
This theorem is referenced by: hashneq0 13193 hashnncl 13195 hash0 13196 hashgt0 13215 hashle00 13226 seqcoll2 13287 prprrab 13293 hashle2pr 13297 hashge2el2difr 13301 ccat0 13394 ccat1st1st 13448 wrdind 13522 wrd2ind 13523 swrdccat3a 13540 swrdccat3blem 13541 rev0 13559 repsw0 13570 cshwidx0 13598 fz1f1o 14485 hashbc0 15756 0hashbc 15758 ram0 15773 cshws0 15855 gsmsymgrfix 17894 sylow1lem1 18059 sylow1lem4 18062 sylow2blem3 18083 frgpnabllem1 18322 0ringnnzr 19317 01eq0ring 19320 vieta1lem2 24111 tgldimor 25442 uhgr0vsize0 26176 uhgr0edgfi 26177 usgr1v0e 26263 fusgrfisbase 26265 vtxd0nedgb 26440 vtxdusgr0edgnelALT 26448 usgrvd0nedg 26485 vtxdginducedm1lem4 26494 finsumvtxdg2size 26502 cyclnspth 26751 iswwlksnx 26788 umgrclwwlkge2 26957 clwwisshclwws 26972 hashecclwwlkn1 27041 umgrhashecclwwlk 27042 vdn0conngrumgrv2 27174 frgrwopreg 27303 frrusgrord0lem 27319 frgrregord013 27382 frgrregord13 27383 frgrogt3nreg 27384 friendshipgt3 27385 hasheuni 30275 signstfvn 30774 signstfveq0a 30781 signshnz 30796 elmrsubrn 31543 lindsrng01 42582 |
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