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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 14056 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) ∈ ℕ0 ∨ (♯‘𝐴) = +∞)) | |
2 | nn0re 12242 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℝ) | |
3 | nn0ge0 12258 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 → 0 ≤ (♯‘𝐴)) | |
4 | ne0gt0 11080 | . . . . . 6 ⊢ (((♯‘𝐴) ∈ ℝ ∧ 0 ≤ (♯‘𝐴)) → ((♯‘𝐴) ≠ 0 ↔ 0 < (♯‘𝐴))) | |
5 | 2, 3, 4 | syl2anc 584 | . . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴) ≠ 0 ↔ 0 < (♯‘𝐴))) |
6 | 5 | bicomd 222 | . . . 4 ⊢ ((♯‘𝐴) ∈ ℕ0 → (0 < (♯‘𝐴) ↔ (♯‘𝐴) ≠ 0)) |
7 | breq2 5078 | . . . . 5 ⊢ ((♯‘𝐴) = +∞ → (0 < (♯‘𝐴) ↔ 0 < +∞)) | |
8 | 0ltpnf 12858 | . . . . . . 7 ⊢ 0 < +∞ | |
9 | 0re 10977 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
10 | renepnf 11023 | . . . . . . . . 9 ⊢ (0 ∈ ℝ → 0 ≠ +∞) | |
11 | 9, 10 | ax-mp 5 | . . . . . . . 8 ⊢ 0 ≠ +∞ |
12 | 11 | necomi 2998 | . . . . . . 7 ⊢ +∞ ≠ 0 |
13 | 8, 12 | 2th 263 | . . . . . 6 ⊢ (0 < +∞ ↔ +∞ ≠ 0) |
14 | neeq1 3006 | . . . . . 6 ⊢ ((♯‘𝐴) = +∞ → ((♯‘𝐴) ≠ 0 ↔ +∞ ≠ 0)) | |
15 | 13, 14 | bitr4id 290 | . . . . 5 ⊢ ((♯‘𝐴) = +∞ → (0 < +∞ ↔ (♯‘𝐴) ≠ 0)) |
16 | 7, 15 | bitrd 278 | . . . 4 ⊢ ((♯‘𝐴) = +∞ → (0 < (♯‘𝐴) ↔ (♯‘𝐴) ≠ 0)) |
17 | 6, 16 | jaoi 854 | . . 3 ⊢ (((♯‘𝐴) ∈ ℕ0 ∨ (♯‘𝐴) = +∞) → (0 < (♯‘𝐴) ↔ (♯‘𝐴) ≠ 0)) |
18 | 1, 17 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (0 < (♯‘𝐴) ↔ (♯‘𝐴) ≠ 0)) |
19 | hasheq0 14078 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) | |
20 | 19 | necon3bid 2988 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) ≠ 0 ↔ 𝐴 ≠ ∅)) |
21 | 18, 20 | bitrd 278 | 1 ⊢ (𝐴 ∈ 𝑉 → (0 < (♯‘𝐴) ↔ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∅c0 4256 class class class wbr 5074 ‘cfv 6433 ℝcr 10870 0cc0 10871 +∞cpnf 11006 < clt 11009 ≤ cle 11010 ℕ0cn0 12233 ♯chash 14044 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-fz 13240 df-hash 14045 |
This theorem is referenced by: hashgt0n0 14080 wrdlenge1n0 14253 ccatws1n0 14342 swrdlsw 14380 pfxsuff1eqwrdeq 14412 ccats1pfxeq 14427 wwlksnextinj 28264 clwwlkext2edg 28420 wwlksext2clwwlk 28421 numclwwlk2lem1lem 28706 tgoldbachgt 32643 lfuhgr2 33080 |
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