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Mirrors > Home > MPE Home > Th. List > hashgt0 | Structured version Visualization version GIF version |
Description: The cardinality of a nonempty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.) |
Ref | Expression |
---|---|
hashgt0 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 0 < (♯‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashge0 13751 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 0 ≤ (♯‘𝐴)) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 0 ≤ (♯‘𝐴)) |
3 | hasheq0 13727 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) | |
4 | 3 | necon3bid 3062 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) ≠ 0 ↔ 𝐴 ≠ ∅)) |
5 | 4 | biimpar 480 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (♯‘𝐴) ≠ 0) |
6 | 2, 5 | jca 514 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (0 ≤ (♯‘𝐴) ∧ (♯‘𝐴) ≠ 0)) |
7 | 0xr 10690 | . . . 4 ⊢ 0 ∈ ℝ* | |
8 | hashxrcl 13721 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ∈ ℝ*) | |
9 | xrltlen 12542 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ (♯‘𝐴) ∈ ℝ*) → (0 < (♯‘𝐴) ↔ (0 ≤ (♯‘𝐴) ∧ (♯‘𝐴) ≠ 0))) | |
10 | 7, 8, 9 | sylancr 589 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (0 < (♯‘𝐴) ↔ (0 ≤ (♯‘𝐴) ∧ (♯‘𝐴) ≠ 0))) |
11 | 10 | biimpar 480 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (0 ≤ (♯‘𝐴) ∧ (♯‘𝐴) ≠ 0)) → 0 < (♯‘𝐴)) |
12 | 6, 11 | syldan 593 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 0 < (♯‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 ≠ wne 3018 ∅c0 4293 class class class wbr 5068 ‘cfv 6357 0cc0 10539 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 ♯chash 13693 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 |
This theorem is referenced by: hashgt0elexb 13766 clwwlkgt0 27766 clwwlkccat 27770 hashxpe 30531 cycpmco2lem5 30774 esumpinfval 31334 |
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