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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 10684 | . . . . . . 7 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 3127 | . . . . . 6 ⊢ ¬ +∞ ∈ ℝ |
3 | hashinf 13698 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | |
4 | 3 | eleq1d 2899 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((♯‘𝐴) ∈ ℝ ↔ +∞ ∈ ℝ)) |
5 | 2, 4 | mtbiri 329 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (♯‘𝐴) ∈ ℝ) |
6 | id 22 | . . . . . 6 ⊢ ((♯‘𝐴) = 0 → (♯‘𝐴) = 0) | |
7 | 0re 10645 | . . . . . 6 ⊢ 0 ∈ ℝ | |
8 | 6, 7 | eqeltrdi 2923 | . . . . 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 8748 | . . . . . . 7 ⊢ ∅ ∈ Fin | |
12 | 10, 11 | eqeltrdi 2923 | . . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin) |
13 | 12 | con3i 157 | . . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ¬ 𝐴 = ∅) |
14 | 13 | adantl 484 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 = ∅) |
15 | 9, 14 | 2falsed 379 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) |
16 | 15 | ex 415 | . 2 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))) |
17 | hashen 13710 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) | |
18 | 11, 17 | mpan2 689 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) |
19 | fz10 12931 | . . . . . 6 ⊢ (1...0) = ∅ | |
20 | 19 | fveq2i 6675 | . . . . 5 ⊢ (♯‘(1...0)) = (♯‘∅) |
21 | 0nn0 11915 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
22 | hashfz1 13709 | . . . . . 6 ⊢ (0 ∈ ℕ0 → (♯‘(1...0)) = 0) | |
23 | 21, 22 | ax-mp 5 | . . . . 5 ⊢ (♯‘(1...0)) = 0 |
24 | 20, 23 | eqtr3i 2848 | . . . 4 ⊢ (♯‘∅) = 0 |
25 | 24 | eqeq2i 2836 | . . 3 ⊢ ((♯‘𝐴) = (♯‘∅) ↔ (♯‘𝐴) = 0) |
26 | en0 8574 | . . 3 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
27 | 18, 25, 26 | 3bitr3g 315 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) |
28 | 16, 27 | pm2.61d2 183 | 1 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∅c0 4293 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ≈ cen 8508 Fincfn 8511 ℝcr 10538 0cc0 10539 1c1 10540 +∞cpnf 10674 ℕ0cn0 11900 ...cfz 12895 ♯chash 13693 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 |
This theorem is referenced by: hashneq0 13728 hashnncl 13730 hash0 13731 hashelne0d 13732 hashgt0 13752 hashle00 13764 seqcoll2 13826 prprrab 13834 hashle2pr 13838 hashge2el2difr 13842 ccat0 13931 ccat1st1st 13986 wrdind 14086 wrd2ind 14087 swrdccat3blem 14103 rev0 14128 repsw0 14141 cshwidx0 14170 fz1f1o 15069 hashbc0 16343 0hashbc 16345 ram0 16360 cshws0 16437 symgvalstruct 18527 gsmsymgrfix 18558 sylow1lem1 18725 sylow1lem4 18728 sylow2blem3 18749 frgpnabllem1 18995 0ringnnzr 20044 01eq0ring 20047 vieta1lem2 24902 tgldimor 26290 uhgr0vsize0 27023 uhgr0edgfi 27024 usgr1v0e 27110 fusgrfisbase 27112 vtxd0nedgb 27272 vtxdusgr0edgnelALT 27280 usgrvd0nedg 27317 vtxdginducedm1lem4 27326 finsumvtxdg2size 27334 cyclnspth 27583 iswwlksnx 27620 umgrclwwlkge2 27771 clwwisshclwws 27795 hashecclwwlkn1 27858 umgrhashecclwwlk 27859 vdn0conngrumgrv2 27977 frgrwopreg 28104 frrusgrord0lem 28120 wlkl0 28148 frgrregord013 28176 frgrregord13 28177 frgrogt3nreg 28178 friendshipgt3 28179 wrdt2ind 30629 tocyc01 30762 lvecdim0i 31006 hasheuni 31346 signstfvn 31841 signstfveq0a 31848 signshnz 31863 spthcycl 32378 usgrgt2cycl 32379 acycgr1v 32398 umgracycusgr 32403 cusgracyclt3v 32405 elmrsubrn 32769 lindsrng01 44530 |
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