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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 11301 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ) | |
2 | ltpnf 11954 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞) | |
3 | rexr 10085 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
4 | pnfxr 10092 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
5 | xrltnle 10105 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐵 < +∞ ↔ ¬ +∞ ≤ 𝐵)) | |
6 | 3, 4, 5 | sylancl 694 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → (𝐵 < +∞ ↔ ¬ +∞ ≤ 𝐵)) |
7 | 2, 6 | mpbid 222 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ → ¬ +∞ ≤ 𝐵) |
8 | 1, 7 | syl 17 | . . . . . 6 ⊢ (𝐵 ∈ ℕ0 → ¬ +∞ ≤ 𝐵) |
9 | hashinf 13122 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞) | |
10 | 9 | breq1d 4663 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((#‘𝐴) ≤ 𝐵 ↔ +∞ ≤ 𝐵)) |
11 | 10 | notbid 308 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (¬ (#‘𝐴) ≤ 𝐵 ↔ ¬ +∞ ≤ 𝐵)) |
12 | 8, 11 | syl5ibrcom 237 | . . . . 5 ⊢ (𝐵 ∈ ℕ0 → ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (#‘𝐴) ≤ 𝐵)) |
13 | 12 | expdimp 453 | . . . 4 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝐴 ∈ 𝑉) → (¬ 𝐴 ∈ Fin → ¬ (#‘𝐴) ≤ 𝐵)) |
14 | 13 | ancoms 469 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ℕ0) → (¬ 𝐴 ∈ Fin → ¬ (#‘𝐴) ≤ 𝐵)) |
15 | 14 | con4d 114 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ℕ0) → ((#‘𝐴) ≤ 𝐵 → 𝐴 ∈ Fin)) |
16 | 15 | 3impia 1261 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ℕ0 ∧ (#‘𝐴) ≤ 𝐵) → 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 Fincfn 7955 ℝcr 9935 +∞cpnf 10071 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 ℕ0cn0 11292 #chash 13117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-hash 13118 |
This theorem is referenced by: 0ringnnzr 19269 fta1glem2 23926 fta1blem 23928 lgsqrlem4 25074 fusgredgfi 26217 idomsubgmo 37776 pgrple2abl 42146 |
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