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Mirrors > Home > MPE Home > Th. List > Mathboxes > rp-isfinite5 | Structured version Visualization version GIF version |
Description: A set is said to be finite if it can be put in one-to-one correspondence with all the natural numbers between 1 and some 𝑛 ∈ ℕ0. (Contributed by RP, 3-Mar-2020.) |
Ref | Expression |
---|---|
rp-isfinite5 | ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6683 | . . . 4 ⊢ (♯‘𝐴) ∈ V | |
2 | hashcl 13718 | . . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
3 | isfinite4 13724 | . . . . . 6 ⊢ (𝐴 ∈ Fin ↔ (1...(♯‘𝐴)) ≈ 𝐴) | |
4 | 3 | biimpi 218 | . . . . 5 ⊢ (𝐴 ∈ Fin → (1...(♯‘𝐴)) ≈ 𝐴) |
5 | 2, 4 | jca 514 | . . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ0 ∧ (1...(♯‘𝐴)) ≈ 𝐴)) |
6 | eleq1 2900 | . . . . . 6 ⊢ (𝑛 = (♯‘𝐴) → (𝑛 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0)) | |
7 | oveq2 7164 | . . . . . . 7 ⊢ (𝑛 = (♯‘𝐴) → (1...𝑛) = (1...(♯‘𝐴))) | |
8 | 7 | breq1d 5076 | . . . . . 6 ⊢ (𝑛 = (♯‘𝐴) → ((1...𝑛) ≈ 𝐴 ↔ (1...(♯‘𝐴)) ≈ 𝐴)) |
9 | 6, 8 | anbi12d 632 | . . . . 5 ⊢ (𝑛 = (♯‘𝐴) → ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((♯‘𝐴) ∈ ℕ0 ∧ (1...(♯‘𝐴)) ≈ 𝐴))) |
10 | 9 | spcegv 3597 | . . . 4 ⊢ ((♯‘𝐴) ∈ V → (((♯‘𝐴) ∈ ℕ0 ∧ (1...(♯‘𝐴)) ≈ 𝐴) → ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))) |
11 | 1, 5, 10 | mpsyl 68 | . . 3 ⊢ (𝐴 ∈ Fin → ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) |
12 | df-rex 3144 | . . 3 ⊢ (∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) | |
13 | 11, 12 | sylibr 236 | . 2 ⊢ (𝐴 ∈ Fin → ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴) |
14 | hasheni 13709 | . . . . . . 7 ⊢ ((1...𝑛) ≈ 𝐴 → (♯‘(1...𝑛)) = (♯‘𝐴)) | |
15 | 14 | eqcomd 2827 | . . . . . 6 ⊢ ((1...𝑛) ≈ 𝐴 → (♯‘𝐴) = (♯‘(1...𝑛))) |
16 | hashfz1 13707 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (♯‘(1...𝑛)) = 𝑛) | |
17 | ovex 7189 | . . . . . . 7 ⊢ (1...(♯‘𝐴)) ∈ V | |
18 | eqtr 2841 | . . . . . . 7 ⊢ (((♯‘𝐴) = (♯‘(1...𝑛)) ∧ (♯‘(1...𝑛)) = 𝑛) → (♯‘𝐴) = 𝑛) | |
19 | oveq2 7164 | . . . . . . . 8 ⊢ ((♯‘𝐴) = 𝑛 → (1...(♯‘𝐴)) = (1...𝑛)) | |
20 | eqeng 8543 | . . . . . . . 8 ⊢ ((1...(♯‘𝐴)) ∈ V → ((1...(♯‘𝐴)) = (1...𝑛) → (1...(♯‘𝐴)) ≈ (1...𝑛))) | |
21 | 19, 20 | syl5 34 | . . . . . . 7 ⊢ ((1...(♯‘𝐴)) ∈ V → ((♯‘𝐴) = 𝑛 → (1...(♯‘𝐴)) ≈ (1...𝑛))) |
22 | 17, 18, 21 | mpsyl 68 | . . . . . 6 ⊢ (((♯‘𝐴) = (♯‘(1...𝑛)) ∧ (♯‘(1...𝑛)) = 𝑛) → (1...(♯‘𝐴)) ≈ (1...𝑛)) |
23 | 15, 16, 22 | syl2anr 598 | . . . . 5 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) → (1...(♯‘𝐴)) ≈ (1...𝑛)) |
24 | entr 8561 | . . . . 5 ⊢ (((1...(♯‘𝐴)) ≈ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → (1...(♯‘𝐴)) ≈ 𝐴) | |
25 | 23, 24 | sylancom 590 | . . . 4 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) → (1...(♯‘𝐴)) ≈ 𝐴) |
26 | 25, 3 | sylibr 236 | . . 3 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) → 𝐴 ∈ Fin) |
27 | 26 | rexlimiva 3281 | . 2 ⊢ (∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴 → 𝐴 ∈ Fin) |
28 | 13, 27 | impbii 211 | 1 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ≈ cen 8506 Fincfn 8509 1c1 10538 ℕ0cn0 11898 ...cfz 12893 ♯chash 13691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 |
This theorem is referenced by: rp-isfinite6 39904 |
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