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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 11300 | . . . . . . 7 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 3046 | . . . . . 6 ⊢ ¬ +∞ ∈ ℝ |
3 | hashinf 14371 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | |
4 | 3 | eleq1d 2824 | . . . . . 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 11261 | . . . . . 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 | 0fi 9081 | . . . . . . 7 ⊢ ∅ ∈ Fin | |
12 | 10, 11 | eqeltrdi 2847 | . . . . . 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 14383 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) | |
18 | 11, 17 | mpan2 691 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘∅) ↔ 𝐴 ≈ ∅)) |
19 | fz10 13582 | . . . . . 6 ⊢ (1...0) = ∅ | |
20 | 19 | fveq2i 6910 | . . . . 5 ⊢ (♯‘(1...0)) = (♯‘∅) |
21 | 0nn0 12539 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
22 | hashfz1 14382 | . . . . . 6 ⊢ (0 ∈ ℕ0 → (♯‘(1...0)) = 0) | |
23 | 21, 22 | ax-mp 5 | . . . . 5 ⊢ (♯‘(1...0)) = 0 |
24 | 20, 23 | eqtr3i 2765 | . . . 4 ⊢ (♯‘∅) = 0 |
25 | 24 | eqeq2i 2748 | . . 3 ⊢ ((♯‘𝐴) = (♯‘∅) ↔ (♯‘𝐴) = 0) |
26 | en0 9057 | . . 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 1537 ∈ wcel 2106 ∅c0 4339 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ≈ cen 8981 Fincfn 8984 ℝcr 11152 0cc0 11153 1c1 11154 +∞cpnf 11290 ℕ0cn0 12524 ...cfz 13544 ♯chash 14366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 |
This theorem is referenced by: hashneq0 14400 hashnncl 14402 hash0 14403 hashelne0d 14404 hashgt0 14424 hashle00 14436 seqcoll2 14501 prprrab 14509 hashle2pr 14513 hashge2el2difr 14517 ccat0 14611 ccat1st1st 14663 wrdind 14757 wrd2ind 14758 swrdccat3blem 14774 rev0 14799 repsw0 14812 cshwidx0 14841 fz1f1o 15743 hashbc0 17039 0hashbc 17041 ram0 17056 cshws0 17136 symgvalstruct 19429 symgvalstructOLD 19430 gsmsymgrfix 19461 sylow1lem1 19631 sylow1lem4 19634 sylow2blem3 19655 frgpnabllem1 19906 0ringnnzr 20542 01eq0ringOLD 20548 vieta1lem2 26368 tgldimor 28525 uhgr0vsize0 29271 uhgr0edgfi 29272 usgr1v0e 29358 fusgrfisbase 29360 vtxd0nedgb 29521 vtxdusgr0edgnelALT 29529 usgrvd0nedg 29566 vtxdginducedm1lem4 29575 finsumvtxdg2size 29583 cyclnspth 29833 iswwlksnx 29870 umgrclwwlkge2 30020 clwwisshclwws 30044 hashecclwwlkn1 30106 umgrhashecclwwlk 30107 vdn0conngrumgrv2 30225 frgrwopreg 30352 frrusgrord0lem 30368 wlkl0 30396 frgrregord013 30424 frgrregord13 30425 frgrogt3nreg 30426 friendshipgt3 30427 wrdt2ind 32923 chnind 32985 chnub 32986 tocyc01 33121 lvecdim0i 33633 hasheuni 34066 signstfvn 34563 signstfveq0a 34570 signshnz 34585 spthcycl 35114 usgrgt2cycl 35115 acycgr1v 35134 umgracycusgr 35139 cusgracyclt3v 35141 elmrsubrn 35505 fsuppind 42577 lindsrng01 48314 |
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