![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ficard | Structured version Visualization version GIF version |
Description: A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
ficard | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 8131 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
2 | carden 9573 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) ↔ 𝐴 ≈ 𝑥)) | |
3 | cardnn 8987 | . . . . . . . 8 ⊢ (𝑥 ∈ ω → (card‘𝑥) = 𝑥) | |
4 | eqtr 2790 | . . . . . . . . 9 ⊢ (((card‘𝐴) = (card‘𝑥) ∧ (card‘𝑥) = 𝑥) → (card‘𝐴) = 𝑥) | |
5 | 4 | expcom 398 | . . . . . . . 8 ⊢ ((card‘𝑥) = 𝑥 → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) = 𝑥)) |
6 | 3, 5 | syl 17 | . . . . . . 7 ⊢ (𝑥 ∈ ω → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) = 𝑥)) |
7 | eleq1a 2845 | . . . . . . 7 ⊢ (𝑥 ∈ ω → ((card‘𝐴) = 𝑥 → (card‘𝐴) ∈ ω)) | |
8 | 6, 7 | syld 47 | . . . . . 6 ⊢ (𝑥 ∈ ω → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) ∈ ω)) |
9 | 8 | adantl 467 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) ∈ ω)) |
10 | 2, 9 | sylbird 250 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → (𝐴 ≈ 𝑥 → (card‘𝐴) ∈ ω)) |
11 | 10 | rexlimdva 3179 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → (card‘𝐴) ∈ ω)) |
12 | 1, 11 | syl5bi 232 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)) |
13 | cardnn 8987 | . . . . . . . 8 ⊢ ((card‘𝐴) ∈ ω → (card‘(card‘𝐴)) = (card‘𝐴)) | |
14 | 13 | eqcomd 2777 | . . . . . . 7 ⊢ ((card‘𝐴) ∈ ω → (card‘𝐴) = (card‘(card‘𝐴))) |
15 | 14 | adantl 467 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (card‘𝐴) ∈ ω) → (card‘𝐴) = (card‘(card‘𝐴))) |
16 | carden 9573 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (card‘𝐴) ∈ ω) → ((card‘𝐴) = (card‘(card‘𝐴)) ↔ 𝐴 ≈ (card‘𝐴))) | |
17 | 15, 16 | mpbid 222 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (card‘𝐴) ∈ ω) → 𝐴 ≈ (card‘𝐴)) |
18 | 17 | ex 397 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ≈ (card‘𝐴))) |
19 | 18 | ancld 540 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ ω → ((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)))) |
20 | breq2 4790 | . . . . 5 ⊢ (𝑥 = (card‘𝐴) → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ (card‘𝐴))) | |
21 | 20 | rspcev 3460 | . . . 4 ⊢ (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
22 | 21, 1 | sylibr 224 | . . 3 ⊢ (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → 𝐴 ∈ Fin) |
23 | 19, 22 | syl6 35 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ∈ Fin)) |
24 | 12, 23 | impbid 202 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∃wrex 3062 class class class wbr 4786 ‘cfv 6029 ωcom 7210 ≈ cen 8104 Fincfn 8107 cardccrd 8959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-ac2 9485 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-isom 6038 df-riota 6752 df-om 7211 df-wrecs 7557 df-recs 7619 df-er 7894 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-card 8963 df-ac 9137 |
This theorem is referenced by: cfpwsdom 9606 |
Copyright terms: Public domain | W3C validator |