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Mirrors > Home > MPE Home > Th. List > hashbnd | Structured version Visualization version GIF version |
Description: If 𝐴 has size bounded by an integer 𝐵, then 𝐴 is finite. (Contributed by Mario Carneiro, 14-Jun-2015.) |
Ref | Expression |
---|---|
hashbnd | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ℕ0 ∧ (♯‘𝐴) ≤ 𝐵) → 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re 11905 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ) | |
2 | ltpnf 12514 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞) | |
3 | rexr 10686 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
4 | pnfxr 10694 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
5 | xrltnle 10707 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐵 < +∞ ↔ ¬ +∞ ≤ 𝐵)) | |
6 | 3, 4, 5 | sylancl 588 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → (𝐵 < +∞ ↔ ¬ +∞ ≤ 𝐵)) |
7 | 2, 6 | mpbid 234 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ → ¬ +∞ ≤ 𝐵) |
8 | 1, 7 | syl 17 | . . . . . 6 ⊢ (𝐵 ∈ ℕ0 → ¬ +∞ ≤ 𝐵) |
9 | hashinf 13694 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | |
10 | 9 | breq1d 5075 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((♯‘𝐴) ≤ 𝐵 ↔ +∞ ≤ 𝐵)) |
11 | 10 | notbid 320 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (¬ (♯‘𝐴) ≤ 𝐵 ↔ ¬ +∞ ≤ 𝐵)) |
12 | 8, 11 | syl5ibrcom 249 | . . . . 5 ⊢ (𝐵 ∈ ℕ0 → ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (♯‘𝐴) ≤ 𝐵)) |
13 | 12 | expdimp 455 | . . . 4 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝐴 ∈ 𝑉) → (¬ 𝐴 ∈ Fin → ¬ (♯‘𝐴) ≤ 𝐵)) |
14 | 13 | ancoms 461 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ℕ0) → (¬ 𝐴 ∈ Fin → ¬ (♯‘𝐴) ≤ 𝐵)) |
15 | 14 | con4d 115 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ℕ0) → ((♯‘𝐴) ≤ 𝐵 → 𝐴 ∈ Fin)) |
16 | 15 | 3impia 1113 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ℕ0 ∧ (♯‘𝐴) ≤ 𝐵) → 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5065 ‘cfv 6354 Fincfn 8508 ℝcr 10535 +∞cpnf 10671 ℝ*cxr 10673 < clt 10674 ≤ cle 10675 ℕ0cn0 11896 ♯chash 13689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-hash 13690 |
This theorem is referenced by: 0ringnnzr 20041 fta1glem2 24759 fta1blem 24761 lgsqrlem4 25924 fusgredgfi 27106 idomsubgmo 39796 pgrple2abl 44412 |
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