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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 8516 | . 2 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
2 | nnon 7566 | . . . 4 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On) | |
3 | ensym 8541 | . . . 4 ⊢ (𝐴 ≈ 𝑥 → 𝑥 ≈ 𝐴) | |
4 | isnumi 9359 | . . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → 𝐴 ∈ dom card) | |
5 | 2, 3, 4 | syl2an 598 | . . 3 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) → 𝐴 ∈ dom card) |
6 | 5 | rexlimiva 3240 | . 2 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ∈ dom card) |
7 | 1, 6 | sylbi 220 | 1 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ∃wrex 3107 class class class wbr 5030 dom cdm 5519 Oncon0 6159 ωcom 7560 ≈ cen 8489 Fincfn 8492 cardccrd 9348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-om 7561 df-er 8272 df-en 8493 df-fin 8496 df-card 9352 |
This theorem is referenced by: ficardom 9374 ficardid 9375 fidomtri 9406 numwdom 9470 fodomfi2 9471 dfac12k 9558 ficardunOLD 9612 ficardun2 9613 ficardun2OLD 9614 pwsdompw 9615 ackbij2 9654 sdom2en01 9713 dfacfin7 9810 fin1a2lem9 9819 domtriomlem 9853 zornn0g 9916 canthnum 10060 pwfseqlem4 10073 uzindi 13345 hashkf 13688 hashgval 13689 hashen 13703 hashdom 13736 symggen 18590 pgpfac1lem5 19194 fiufl 22521 finixpnum 35042 poimirlem32 35089 ttac 39977 |
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