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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 14331 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) ∈ ℕ0 ∨ (♯‘𝐴) = +∞)) | |
2 | nn0re 12509 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℝ) | |
3 | nn0ge0 12525 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 → 0 ≤ (♯‘𝐴)) | |
4 | ne0gt0 11347 | . . . . . 6 ⊢ (((♯‘𝐴) ∈ ℝ ∧ 0 ≤ (♯‘𝐴)) → ((♯‘𝐴) ≠ 0 ↔ 0 < (♯‘𝐴))) | |
5 | 2, 3, 4 | syl2anc 582 | . . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴) ≠ 0 ↔ 0 < (♯‘𝐴))) |
6 | 5 | bicomd 222 | . . . 4 ⊢ ((♯‘𝐴) ∈ ℕ0 → (0 < (♯‘𝐴) ↔ (♯‘𝐴) ≠ 0)) |
7 | breq2 5145 | . . . . 5 ⊢ ((♯‘𝐴) = +∞ → (0 < (♯‘𝐴) ↔ 0 < +∞)) | |
8 | 0ltpnf 13132 | . . . . . . 7 ⊢ 0 < +∞ | |
9 | 0re 11244 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
10 | renepnf 11290 | . . . . . . . . 9 ⊢ (0 ∈ ℝ → 0 ≠ +∞) | |
11 | 9, 10 | ax-mp 5 | . . . . . . . 8 ⊢ 0 ≠ +∞ |
12 | 11 | necomi 2985 | . . . . . . 7 ⊢ +∞ ≠ 0 |
13 | 8, 12 | 2th 263 | . . . . . 6 ⊢ (0 < +∞ ↔ +∞ ≠ 0) |
14 | neeq1 2993 | . . . . . 6 ⊢ ((♯‘𝐴) = +∞ → ((♯‘𝐴) ≠ 0 ↔ +∞ ≠ 0)) | |
15 | 13, 14 | bitr4id 289 | . . . . 5 ⊢ ((♯‘𝐴) = +∞ → (0 < +∞ ↔ (♯‘𝐴) ≠ 0)) |
16 | 7, 15 | bitrd 278 | . . . 4 ⊢ ((♯‘𝐴) = +∞ → (0 < (♯‘𝐴) ↔ (♯‘𝐴) ≠ 0)) |
17 | 6, 16 | jaoi 855 | . . 3 ⊢ (((♯‘𝐴) ∈ ℕ0 ∨ (♯‘𝐴) = +∞) → (0 < (♯‘𝐴) ↔ (♯‘𝐴) ≠ 0)) |
18 | 1, 17 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (0 < (♯‘𝐴) ↔ (♯‘𝐴) ≠ 0)) |
19 | hasheq0 14352 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) | |
20 | 19 | necon3bid 2975 | . 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 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∅c0 4316 class class class wbr 5141 ‘cfv 6541 ℝcr 11135 0cc0 11136 +∞cpnf 11273 < clt 11276 ≤ cle 11277 ℕ0cn0 12500 ♯chash 14319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-n0 12501 df-xnn0 12573 df-z 12587 df-uz 12851 df-fz 13515 df-hash 14320 |
This theorem is referenced by: hashgt0n0 14354 wrdlenge1n0 14530 ccatws1n0 14612 swrdlsw 14647 pfxsuff1eqwrdeq 14679 ccats1pfxeq 14694 wwlksnextinj 29726 clwwlkext2edg 29882 wwlksext2clwwlk 29883 numclwwlk2lem1lem 30168 tgoldbachgt 34324 lfuhgr2 34757 |
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