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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 483 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin) | |
2 | simplr 767 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐴 ∈ Fin) → 𝐴 ≠ ∅) | |
3 | hashnncl 14365 | . . . . 5 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅)) | |
4 | 3 | biimpar 476 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (♯‘𝐴) ∈ ℕ) |
5 | 1, 2, 4 | syl2anc 582 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐴 ∈ Fin) → (♯‘𝐴) ∈ ℕ) |
6 | 5 | nnge1d 12298 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐴 ∈ Fin) → 1 ≤ (♯‘𝐴)) |
7 | 1xr 11311 | . . . 4 ⊢ 1 ∈ ℝ* | |
8 | pnfge 13150 | . . . 4 ⊢ (1 ∈ ℝ* → 1 ≤ +∞) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ 1 ≤ +∞ |
10 | hashinf 14334 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | |
11 | 10 | adantlr 713 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) |
12 | 9, 11 | breqtrrid 5190 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ ¬ 𝐴 ∈ Fin) → 1 ≤ (♯‘𝐴)) |
13 | 6, 12 | pm2.61dan 811 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 1 ≤ (♯‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∅c0 4326 class class class wbr 5152 ‘cfv 6553 Fincfn 8970 1c1 11147 +∞cpnf 11283 ℝ*cxr 11285 ≤ cle 11287 ℕcn 12250 ♯chash 14329 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-hash 14330 |
This theorem is referenced by: 1elfz0hash 14389 hashnn0n0nn 14390 wlk1walk 29473 wlkiswwlks2 29706 friendship 30229 hasheuni 33737 aks6d1c6lem3 41676 fourierdlem52 45575 |
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