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Mirrors > Home > MPE Home > Th. List > hashge1 | Structured version Visualization version GIF version |
Description: The cardinality of a nonempty set is greater than or equal to one. (Contributed by Thierry Arnoux, 20-Jun-2017.) |
Ref | Expression |
---|---|
hashge1 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 1 ≤ (♯‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 478 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin) | |
2 | simplr 786 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐴 ∈ Fin) → 𝐴 ≠ ∅) | |
3 | hashnncl 13407 | . . . . 5 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅)) | |
4 | 3 | biimpar 470 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (♯‘𝐴) ∈ ℕ) |
5 | 1, 2, 4 | syl2anc 580 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐴 ∈ Fin) → (♯‘𝐴) ∈ ℕ) |
6 | 5 | nnge1d 11361 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐴 ∈ Fin) → 1 ≤ (♯‘𝐴)) |
7 | 1re 10328 | . . . . 5 ⊢ 1 ∈ ℝ | |
8 | 7 | rexri 10387 | . . . 4 ⊢ 1 ∈ ℝ* |
9 | pnfge 12211 | . . . 4 ⊢ (1 ∈ ℝ* → 1 ≤ +∞) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ 1 ≤ +∞ |
11 | hashinf 13375 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | |
12 | 11 | adantlr 707 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) |
13 | 10, 12 | syl5breqr 4881 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ ¬ 𝐴 ∈ Fin) → 1 ≤ (♯‘𝐴)) |
14 | 6, 13 | pm2.61dan 848 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 1 ≤ (♯‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ∅c0 4115 class class class wbr 4843 ‘cfv 6101 Fincfn 8195 1c1 10225 +∞cpnf 10360 ℝ*cxr 10362 ≤ cle 10364 ℕcn 11312 ♯chash 13370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-hash 13371 |
This theorem is referenced by: 1elfz0hash 13429 hashnn0n0nn 13430 wlk1walk 26888 wlkiswwlks2 27132 friendship 27784 hasheuni 30663 fourierdlem52 41118 |
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