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| Mirrors > Home > MPE Home > Th. List > hasheq0 | Structured version Visualization version GIF version | ||
| Description: Two ways of saying a 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 11302 | . . . . . . 7 ⊢ +∞ ∉ ℝ | |
| 2 | 1 | neli 3048 | . . . . . 6 ⊢ ¬ +∞ ∈ ℝ |
| 3 | hashinf 14374 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | |
| 4 | 3 | eleq1d 2826 | . . . . . 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 11263 | . . . . . 6 ⊢ 0 ∈ ℝ | |
| 8 | 6, 7 | eqeltrdi 2849 | . . . . 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 9082 | . . . . . . 7 ⊢ ∅ ∈ Fin | |
| 12 | 10, 11 | eqeltrdi 2849 | . . . . . 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 14386 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) | |
| 18 | 11, 17 | mpan2 691 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) |
| 19 | fz10 13585 | . . . . . 6 ⊢ (1...0) = ∅ | |
| 20 | 19 | fveq2i 6909 | . . . . 5 ⊢ (♯‘(1...0)) = (♯‘∅) |
| 21 | 0nn0 12541 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
| 22 | hashfz1 14385 | . . . . . 6 ⊢ (0 ∈ ℕ0 → (♯‘(1...0)) = 0) | |
| 23 | 21, 22 | ax-mp 5 | . . . . 5 ⊢ (♯‘(1...0)) = 0 |
| 24 | 20, 23 | eqtr3i 2767 | . . . 4 ⊢ (♯‘∅) = 0 |
| 25 | 24 | eqeq2i 2750 | . . 3 ⊢ ((♯‘𝐴) = (♯‘∅) ↔ (♯‘𝐴) = 0) |
| 26 | en0 9058 | . . 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 2108 ∅c0 4333 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ≈ cen 8982 Fincfn 8985 ℝcr 11154 0cc0 11155 1c1 11156 +∞cpnf 11292 ℕ0cn0 12526 ...cfz 13547 ♯chash 14369 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 |
| This theorem is referenced by: hashneq0 14403 hashnncl 14405 hash0 14406 hashelne0d 14407 hashgt0 14427 hashle00 14439 seqcoll2 14504 prprrab 14512 hashle2pr 14516 hashge2el2difr 14520 ccat0 14614 ccat1st1st 14666 wrdind 14760 wrd2ind 14761 swrdccat3blem 14777 rev0 14802 repsw0 14815 cshwidx0 14844 fz1f1o 15746 hashbc0 17043 0hashbc 17045 ram0 17060 cshws0 17139 symgvalstruct 19414 symgvalstructOLD 19415 gsmsymgrfix 19446 sylow1lem1 19616 sylow1lem4 19619 sylow2blem3 19640 frgpnabllem1 19891 0ringnnzr 20525 01eq0ringOLD 20531 vieta1lem2 26353 tgldimor 28510 uhgr0vsize0 29256 uhgr0edgfi 29257 usgr1v0e 29343 fusgrfisbase 29345 vtxd0nedgb 29506 vtxdusgr0edgnelALT 29514 usgrvd0nedg 29551 vtxdginducedm1lem4 29560 finsumvtxdg2size 29568 cyclnspth 29821 iswwlksnx 29860 umgrclwwlkge2 30010 clwwisshclwws 30034 hashecclwwlkn1 30096 umgrhashecclwwlk 30097 vdn0conngrumgrv2 30215 frgrwopreg 30342 frrusgrord0lem 30358 wlkl0 30386 frgrregord013 30414 frgrregord13 30415 frgrogt3nreg 30416 friendshipgt3 30417 wrdt2ind 32938 chnind 33001 chnub 33002 tocyc01 33138 lvecdim0i 33656 hasheuni 34086 signstfvn 34584 signstfveq0a 34591 signshnz 34606 spthcycl 35134 usgrgt2cycl 35135 acycgr1v 35154 umgracycusgr 35159 cusgracyclt3v 35161 elmrsubrn 35525 fsuppind 42600 lindsrng01 48385 |
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