| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > finnum | Structured version Visualization version GIF version | ||
| Description: Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| Ref | Expression |
|---|---|
| finnum | ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isfi 9016 | . 2 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
| 2 | nnon 7893 | . . . 4 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On) | |
| 3 | ensym 9043 | . . . 4 ⊢ (𝐴 ≈ 𝑥 → 𝑥 ≈ 𝐴) | |
| 4 | isnumi 9986 | . . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → 𝐴 ∈ dom card) | |
| 5 | 2, 3, 4 | syl2an 596 | . . 3 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) → 𝐴 ∈ dom card) |
| 6 | 5 | rexlimiva 3147 | . 2 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ∈ dom card) |
| 7 | 1, 6 | sylbi 217 | 1 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∃wrex 3070 class class class wbr 5143 dom cdm 5685 Oncon0 6384 ωcom 7887 ≈ cen 8982 Fincfn 8985 cardccrd 9975 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-om 7888 df-er 8745 df-en 8986 df-fin 8989 df-card 9979 |
| This theorem is referenced by: ficardom 10001 ficardid 10002 fidomtri 10033 numwdom 10099 fodomfi2 10100 dfac12k 10188 ficardun2 10242 pwsdompw 10243 ackbij2 10282 sdom2en01 10342 dfacfin7 10439 fin1a2lem9 10448 domtriomlem 10482 zornn0g 10545 canthnum 10689 pwfseqlem4 10702 uzindi 14023 hashkf 14371 hashgval 14372 hashen 14386 hashdom 14418 symggen 19488 pgpfac1lem5 20099 fiufl 23924 fineqvacALT 35112 finixpnum 37612 poimirlem32 37659 ttac 43048 |
| Copyright terms: Public domain | W3C validator |