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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 11178 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ) | |
2 | ltpnf 11830 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞) | |
3 | rexr 9964 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
4 | pnfxr 9971 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
5 | xrltnle 9984 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐵 < +∞ ↔ ¬ +∞ ≤ 𝐵)) | |
6 | 3, 4, 5 | sylancl 693 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → (𝐵 < +∞ ↔ ¬ +∞ ≤ 𝐵)) |
7 | 2, 6 | mpbid 221 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ → ¬ +∞ ≤ 𝐵) |
8 | 1, 7 | syl 17 | . . . . . 6 ⊢ (𝐵 ∈ ℕ0 → ¬ +∞ ≤ 𝐵) |
9 | hashinf 12984 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞) | |
10 | 9 | breq1d 4593 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((#‘𝐴) ≤ 𝐵 ↔ +∞ ≤ 𝐵)) |
11 | 10 | notbid 307 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (¬ (#‘𝐴) ≤ 𝐵 ↔ ¬ +∞ ≤ 𝐵)) |
12 | 8, 11 | syl5ibrcom 236 | . . . . 5 ⊢ (𝐵 ∈ ℕ0 → ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (#‘𝐴) ≤ 𝐵)) |
13 | 12 | expdimp 452 | . . . 4 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝐴 ∈ 𝑉) → (¬ 𝐴 ∈ Fin → ¬ (#‘𝐴) ≤ 𝐵)) |
14 | 13 | ancoms 468 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ℕ0) → (¬ 𝐴 ∈ Fin → ¬ (#‘𝐴) ≤ 𝐵)) |
15 | 14 | con4d 113 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ℕ0) → ((#‘𝐴) ≤ 𝐵 → 𝐴 ∈ Fin)) |
16 | 15 | 3impia 1253 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ℕ0 ∧ (#‘𝐴) ≤ 𝐵) → 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 Fincfn 7841 ℝcr 9814 +∞cpnf 9950 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 ℕ0cn0 11169 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-hash 12980 |
This theorem is referenced by: 0ringnnzr 19090 fta1glem2 23730 fta1blem 23732 lgsqrlem4 24874 upgredg 25811 fiusgraedgfi 25936 idomsubgmo 36795 fusgredgfi 40544 pgrple2abl 41940 |
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