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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 9016 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
| 2 | 1 | biimpi 216 | . 2 ⊢ (𝐴 ∈ Fin → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
| 3 | finnum 9988 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) | |
| 4 | cardid2 9993 | . . . . . . . 8 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
| 5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴) |
| 6 | entr 9046 | . . . . . . 7 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ 𝐴 ≈ 𝑥) → (card‘𝐴) ≈ 𝑥) | |
| 7 | 5, 6 | sylan 580 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝑥) → (card‘𝐴) ≈ 𝑥) |
| 8 | cardon 9984 | . . . . . . 7 ⊢ (card‘𝐴) ∈ On | |
| 9 | onomeneq 9265 | . . . . . . 7 ⊢ (((card‘𝐴) ∈ On ∧ 𝑥 ∈ ω) → ((card‘𝐴) ≈ 𝑥 ↔ (card‘𝐴) = 𝑥)) | |
| 10 | 8, 9 | mpan 690 | . . . . . 6 ⊢ (𝑥 ∈ ω → ((card‘𝐴) ≈ 𝑥 ↔ (card‘𝐴) = 𝑥)) |
| 11 | 7, 10 | imbitrid 244 | . . . . 5 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝑥) → (card‘𝐴) = 𝑥)) |
| 12 | eleq1a 2836 | . . . . 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 416 | . . 3 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω))) |
| 15 | 14 | rexlimiv 3148 | . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 class class class wbr 5143 dom cdm 5685 Oncon0 6384 ‘cfv 6561 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-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 |
| This theorem is referenced by: cardnn 10003 isinffi 10032 finnisoeu 10153 iunfictbso 10154 ficardadju 10240 ficardun 10241 ficardun2 10242 pwsdompw 10243 ackbij1lem5 10263 ackbij1lem9 10267 ackbij1lem10 10268 ackbij1lem14 10272 ackbij1b 10278 ackbij2lem2 10279 ackbij2 10282 fin23lem22 10367 fin1a2lem11 10450 domtriomlem 10482 pwfseqlem4a 10701 pwfseqlem4 10702 hashkf 14371 hashginv 14373 hashcard 14394 hashcl 14395 hashdom 14418 hashun 14421 ishashinf 14502 ackbijnn 15864 mreexexd 17691 |
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