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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 8923 | . 2 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
2 | nnon 7813 | . . . 4 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On) | |
3 | ensym 8950 | . . . 4 ⊢ (𝐴 ≈ 𝑥 → 𝑥 ≈ 𝐴) | |
4 | isnumi 9889 | . . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → 𝐴 ∈ dom card) | |
5 | 2, 3, 4 | syl2an 597 | . . 3 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) → 𝐴 ∈ dom card) |
6 | 5 | rexlimiva 3145 | . 2 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ∈ dom card) |
7 | 1, 6 | sylbi 216 | 1 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∃wrex 3074 class class class wbr 5110 dom cdm 5638 Oncon0 6322 ωcom 7807 ≈ cen 8887 Fincfn 8890 cardccrd 9878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-ord 6325 df-on 6326 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-om 7808 df-er 8655 df-en 8891 df-fin 8894 df-card 9882 |
This theorem is referenced by: ficardom 9904 ficardid 9905 fidomtri 9936 numwdom 10002 fodomfi2 10003 dfac12k 10090 ficardunOLD 10144 ficardun2 10145 ficardun2OLD 10146 pwsdompw 10147 ackbij2 10186 sdom2en01 10245 dfacfin7 10342 fin1a2lem9 10351 domtriomlem 10385 zornn0g 10448 canthnum 10592 pwfseqlem4 10605 uzindi 13894 hashkf 14239 hashgval 14240 hashen 14254 hashdom 14286 symggen 19259 pgpfac1lem5 19865 fiufl 23283 fineqvacALT 33739 finixpnum 36092 poimirlem32 36139 ttac 41389 |
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