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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 8976 | . 2 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
2 | nnon 7865 | . . . 4 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On) | |
3 | ensym 9003 | . . . 4 ⊢ (𝐴 ≈ 𝑥 → 𝑥 ≈ 𝐴) | |
4 | isnumi 9945 | . . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → 𝐴 ∈ dom card) | |
5 | 2, 3, 4 | syl2an 594 | . . 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 2104 ∃wrex 3068 class class class wbr 5149 dom cdm 5677 Oncon0 6365 ωcom 7859 ≈ cen 8940 Fincfn 8943 cardccrd 9934 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-om 7860 df-er 8707 df-en 8944 df-fin 8947 df-card 9938 |
This theorem is referenced by: ficardom 9960 ficardid 9961 fidomtri 9992 numwdom 10058 fodomfi2 10059 dfac12k 10146 ficardunOLD 10200 ficardun2 10201 ficardun2OLD 10202 pwsdompw 10203 ackbij2 10242 sdom2en01 10301 dfacfin7 10398 fin1a2lem9 10407 domtriomlem 10441 zornn0g 10504 canthnum 10648 pwfseqlem4 10661 uzindi 13953 hashkf 14298 hashgval 14299 hashen 14313 hashdom 14345 symggen 19381 pgpfac1lem5 19992 fiufl 23642 fineqvacALT 34394 finixpnum 36778 poimirlem32 36825 ttac 42079 |
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