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Mirrors > Home > MPE Home > Th. List > ficardom | Structured version Visualization version GIF version |
Description: The cardinal number of a finite set is a finite ordinal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 4-Feb-2013.) |
Ref | Expression |
---|---|
ficardom | ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 8912 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
2 | 1 | biimpi 215 | . 2 ⊢ (𝐴 ∈ Fin → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
3 | finnum 9880 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) | |
4 | cardid2 9885 | . . . . . . . 8 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴) |
6 | entr 8942 | . . . . . . 7 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ 𝐴 ≈ 𝑥) → (card‘𝐴) ≈ 𝑥) | |
7 | 5, 6 | sylan 580 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝑥) → (card‘𝐴) ≈ 𝑥) |
8 | cardon 9876 | . . . . . . 7 ⊢ (card‘𝐴) ∈ On | |
9 | onomeneq 9168 | . . . . . . 7 ⊢ (((card‘𝐴) ∈ On ∧ 𝑥 ∈ ω) → ((card‘𝐴) ≈ 𝑥 ↔ (card‘𝐴) = 𝑥)) | |
10 | 8, 9 | mpan 688 | . . . . . 6 ⊢ (𝑥 ∈ ω → ((card‘𝐴) ≈ 𝑥 ↔ (card‘𝐴) = 𝑥)) |
11 | 7, 10 | imbitrid 243 | . . . . 5 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝑥) → (card‘𝐴) = 𝑥)) |
12 | eleq1a 2833 | . . . . 5 ⊢ (𝑥 ∈ ω → ((card‘𝐴) = 𝑥 → (card‘𝐴) ∈ ω)) | |
13 | 11, 12 | syld 47 | . . . 4 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝑥) → (card‘𝐴) ∈ ω)) |
14 | 13 | expcomd 417 | . . 3 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω))) |
15 | 14 | rexlimiv 3143 | . 2 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)) |
16 | 2, 15 | mpcom 38 | 1 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3071 class class class wbr 5103 dom cdm 5631 Oncon0 6315 ‘cfv 6493 ωcom 7798 ≈ cen 8876 Fincfn 8879 cardccrd 9867 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-om 7799 df-1o 8408 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-card 9871 |
This theorem is referenced by: cardnn 9895 isinffi 9924 finnisoeu 10045 iunfictbso 10046 ficardadju 10131 ficardun 10132 ficardunOLD 10133 ficardun2 10134 ficardun2OLD 10135 pwsdompw 10136 ackbij1lem5 10156 ackbij1lem9 10160 ackbij1lem10 10161 ackbij1lem14 10165 ackbij1b 10171 ackbij2lem2 10172 ackbij2 10175 fin23lem22 10259 fin1a2lem11 10342 domtriomlem 10374 pwfseqlem4a 10593 pwfseqlem4 10594 hashkf 14224 hashginv 14226 hashcard 14247 hashcl 14248 hashdom 14271 hashun 14274 ishashinf 14354 ackbijnn 15705 mreexexd 17520 |
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