![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > hashneq0 | Structured version Visualization version GIF version |
Description: Two ways of saying a set is not empty. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
Ref | Expression |
---|---|
hashneq0 | ⊢ (𝐴 ∈ 𝑉 → (0 < (♯‘𝐴) ↔ 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashnn0pnf 14249 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) ∈ ℕ0 ∨ (♯‘𝐴) = +∞)) | |
2 | nn0re 12429 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℝ) | |
3 | nn0ge0 12445 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 → 0 ≤ (♯‘𝐴)) | |
4 | ne0gt0 11267 | . . . . . 6 ⊢ (((♯‘𝐴) ∈ ℝ ∧ 0 ≤ (♯‘𝐴)) → ((♯‘𝐴) ≠ 0 ↔ 0 < (♯‘𝐴))) | |
5 | 2, 3, 4 | syl2anc 585 | . . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴) ≠ 0 ↔ 0 < (♯‘𝐴))) |
6 | 5 | bicomd 222 | . . . 4 ⊢ ((♯‘𝐴) ∈ ℕ0 → (0 < (♯‘𝐴) ↔ (♯‘𝐴) ≠ 0)) |
7 | breq2 5114 | . . . . 5 ⊢ ((♯‘𝐴) = +∞ → (0 < (♯‘𝐴) ↔ 0 < +∞)) | |
8 | 0ltpnf 13050 | . . . . . . 7 ⊢ 0 < +∞ | |
9 | 0re 11164 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
10 | renepnf 11210 | . . . . . . . . 9 ⊢ (0 ∈ ℝ → 0 ≠ +∞) | |
11 | 9, 10 | ax-mp 5 | . . . . . . . 8 ⊢ 0 ≠ +∞ |
12 | 11 | necomi 2999 | . . . . . . 7 ⊢ +∞ ≠ 0 |
13 | 8, 12 | 2th 264 | . . . . . 6 ⊢ (0 < +∞ ↔ +∞ ≠ 0) |
14 | neeq1 3007 | . . . . . 6 ⊢ ((♯‘𝐴) = +∞ → ((♯‘𝐴) ≠ 0 ↔ +∞ ≠ 0)) | |
15 | 13, 14 | bitr4id 290 | . . . . 5 ⊢ ((♯‘𝐴) = +∞ → (0 < +∞ ↔ (♯‘𝐴) ≠ 0)) |
16 | 7, 15 | bitrd 279 | . . . 4 ⊢ ((♯‘𝐴) = +∞ → (0 < (♯‘𝐴) ↔ (♯‘𝐴) ≠ 0)) |
17 | 6, 16 | jaoi 856 | . . 3 ⊢ (((♯‘𝐴) ∈ ℕ0 ∨ (♯‘𝐴) = +∞) → (0 < (♯‘𝐴) ↔ (♯‘𝐴) ≠ 0)) |
18 | 1, 17 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (0 < (♯‘𝐴) ↔ (♯‘𝐴) ≠ 0)) |
19 | hasheq0 14270 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) | |
20 | 19 | necon3bid 2989 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) ≠ 0 ↔ 𝐴 ≠ ∅)) |
21 | 18, 20 | bitrd 279 | 1 ⊢ (𝐴 ∈ 𝑉 → (0 < (♯‘𝐴) ↔ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∅c0 4287 class class class wbr 5110 ‘cfv 6501 ℝcr 11057 0cc0 11058 +∞cpnf 11193 < clt 11196 ≤ cle 11197 ℕ0cn0 12420 ♯chash 14237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-fz 13432 df-hash 14238 |
This theorem is referenced by: hashgt0n0 14272 wrdlenge1n0 14445 ccatws1n0 14527 swrdlsw 14562 pfxsuff1eqwrdeq 14594 ccats1pfxeq 14609 wwlksnextinj 28886 clwwlkext2edg 29042 wwlksext2clwwlk 29043 numclwwlk2lem1lem 29328 tgoldbachgt 33316 lfuhgr2 33752 |
Copyright terms: Public domain | W3C validator |