Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 9960 | . . . . . . 7 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 2885 | . . . . . 6 ⊢ ¬ +∞ ∈ ℝ |
3 | hashinf 12984 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞) | |
4 | 3 | eleq1d 2672 | . . . . . 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 9919 | . . . . . 6 ⊢ 0 ∈ ℝ | |
8 | 6, 7 | syl6eqel 2696 | . . . . 5 ⊢ ((#‘𝐴) = 0 → (#‘𝐴) ∈ ℝ) |
9 | 5, 8 | nsyl 134 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (#‘𝐴) = 0) |
10 | id 22 | . . . . . . 7 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
11 | 0fin 8073 | . . . . . . 7 ⊢ ∅ ∈ Fin | |
12 | 10, 11 | syl6eqel 2696 | . . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin) |
13 | 12 | con3i 149 | . . . . 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 12997 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∅ ∈ Fin) → ((#‘𝐴) = (#‘∅) ↔ 𝐴 ≈ ∅)) | |
18 | 11, 17 | mpan2 703 | . . 3 ⊢ (𝐴 ∈ Fin → ((#‘𝐴) = (#‘∅) ↔ 𝐴 ≈ ∅)) |
19 | fz10 12233 | . . . . . 6 ⊢ (1...0) = ∅ | |
20 | 19 | fveq2i 6106 | . . . . 5 ⊢ (#‘(1...0)) = (#‘∅) |
21 | 0nn0 11184 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
22 | hashfz1 12996 | . . . . . 6 ⊢ (0 ∈ ℕ0 → (#‘(1...0)) = 0) | |
23 | 21, 22 | ax-mp 5 | . . . . 5 ⊢ (#‘(1...0)) = 0 |
24 | 20, 23 | eqtr3i 2634 | . . . 4 ⊢ (#‘∅) = 0 |
25 | 24 | eqeq2i 2622 | . . 3 ⊢ ((#‘𝐴) = (#‘∅) ↔ (#‘𝐴) = 0) |
26 | en0 7905 | . . 3 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
27 | 18, 25, 26 | 3bitr3g 301 | . 2 ⊢ (𝐴 ∈ Fin → ((#‘𝐴) = 0 ↔ 𝐴 = ∅)) |
28 | 16, 27 | pm2.61d2 171 | 1 ⊢ (𝐴 ∈ 𝑉 → ((#‘𝐴) = 0 ↔ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∅c0 3874 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ≈ cen 7838 Fincfn 7841 ℝcr 9814 0cc0 9815 1c1 9816 +∞cpnf 9950 ℕ0cn0 11169 ...cfz 12197 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 |
This theorem is referenced by: hashneq0 13016 hashnncl 13018 hash0 13019 hashgt0 13038 hashle00 13049 seqcoll2 13106 prprrab 13112 hashge2el2difr 13118 wrdind 13328 wrd2ind 13329 swrdccat3a 13345 swrdccat3blem 13346 rev0 13364 repsw0 13375 cshwidx0 13403 fz1f1o 14288 hashbc0 15547 0hashbc 15549 ram0 15564 cshws0 15646 gsmsymgrfix 17671 sylow1lem1 17836 sylow1lem4 17839 sylow2blem3 17860 frgpnabllem1 18099 0ringnnzr 19090 01eq0ring 19093 vieta1lem2 23870 tgldimor 25197 upgredg 25811 isusgra0 25876 usgraop 25879 usgrafisindb0 25937 wwlkn0s 26233 clwwlkgt0 26299 hashecclwwlkn1 26361 usghashecclwwlk 26362 vdusgra0nedg 26435 usgravd0nedg 26445 vdn0frgrav2 26550 vdgn0frgrav2 26551 frgrawopreg 26576 frgregordn0 26597 frgrareg 26644 frgraregord013 26645 frgraregord13 26646 frgraogt3nreg 26647 friendshipgt3 26648 hasheuni 29474 signstfvn 29972 signstfveq0a 29979 signshnz 29994 elmrsubrn 30671 uhgr0vsize0 40465 uhgr0edgfi 40466 usgr1v0e 40545 fusgrfisbase 40547 vtxd0nedgb 40703 vtxdusgr0edgnelALT 40711 usgrvd0nedg 40749 cyclnsPth 41006 iswwlksnx 41042 isclwwlksnx 41197 umgrclwwlksge2 41219 clwwisshclwws 41235 hashecclwwlksn1 41261 umgrhashecclwwlk 41262 vdn0conngrumgrv2 41363 frgrwopreg 41486 frrusgrord0 41503 av-frgraregord013 41549 av-frgraregord13 41550 av-frgraogt3nreg 41551 av-friendshipgt3 41552 lindsrng01 42051 |
Copyright terms: Public domain | W3C validator |