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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 14309 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) ∈ ℕ0 ∨ (♯‘𝐴) = +∞)) | |
2 | nn0re 12488 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℝ) | |
3 | nn0ge0 12504 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 → 0 ≤ (♯‘𝐴)) | |
4 | ne0gt0 11326 | . . . . . 6 ⊢ (((♯‘𝐴) ∈ ℝ ∧ 0 ≤ (♯‘𝐴)) → ((♯‘𝐴) ≠ 0 ↔ 0 < (♯‘𝐴))) | |
5 | 2, 3, 4 | syl2anc 583 | . . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴) ≠ 0 ↔ 0 < (♯‘𝐴))) |
6 | 5 | bicomd 222 | . . . 4 ⊢ ((♯‘𝐴) ∈ ℕ0 → (0 < (♯‘𝐴) ↔ (♯‘𝐴) ≠ 0)) |
7 | breq2 5152 | . . . . 5 ⊢ ((♯‘𝐴) = +∞ → (0 < (♯‘𝐴) ↔ 0 < +∞)) | |
8 | 0ltpnf 13109 | . . . . . . 7 ⊢ 0 < +∞ | |
9 | 0re 11223 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
10 | renepnf 11269 | . . . . . . . . 9 ⊢ (0 ∈ ℝ → 0 ≠ +∞) | |
11 | 9, 10 | ax-mp 5 | . . . . . . . 8 ⊢ 0 ≠ +∞ |
12 | 11 | necomi 2994 | . . . . . . 7 ⊢ +∞ ≠ 0 |
13 | 8, 12 | 2th 264 | . . . . . 6 ⊢ (0 < +∞ ↔ +∞ ≠ 0) |
14 | neeq1 3002 | . . . . . 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 854 | . . 3 ⊢ (((♯‘𝐴) ∈ ℕ0 ∨ (♯‘𝐴) = +∞) → (0 < (♯‘𝐴) ↔ (♯‘𝐴) ≠ 0)) |
18 | 1, 17 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (0 < (♯‘𝐴) ↔ (♯‘𝐴) ≠ 0)) |
19 | hasheq0 14330 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) | |
20 | 19 | necon3bid 2984 | . 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 844 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∅c0 4322 class class class wbr 5148 ‘cfv 6543 ℝcr 11115 0cc0 11116 +∞cpnf 11252 < clt 11255 ≤ cle 11256 ℕ0cn0 12479 ♯chash 14297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-xnn0 12552 df-z 12566 df-uz 12830 df-fz 13492 df-hash 14298 |
This theorem is referenced by: hashgt0n0 14332 wrdlenge1n0 14507 ccatws1n0 14589 swrdlsw 14624 pfxsuff1eqwrdeq 14656 ccats1pfxeq 14671 wwlksnextinj 29587 clwwlkext2edg 29743 wwlksext2clwwlk 29744 numclwwlk2lem1lem 30029 tgoldbachgt 34140 lfuhgr2 34574 |
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