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| Mirrors > Home > MPE Home > Th. List > cardnn | Structured version Visualization version GIF version | ||
| Description: The cardinality of a natural number is the number. Corollary 10.23 of [TakeutiZaring] p. 90. (Contributed by Mario Carneiro, 7-Jan-2013.) |
| Ref | Expression |
|---|---|
| cardnn | ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nnon 7893 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
| 2 | onenon 9989 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | |
| 3 | cardid2 9993 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
| 4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝐴 ∈ ω → (card‘𝐴) ≈ 𝐴) |
| 5 | nnfi 9207 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) | |
| 6 | ficardom 10001 | . . . 4 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω) | |
| 7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ ω → (card‘𝐴) ∈ ω) |
| 8 | nneneq 9246 | . . 3 ⊢ (((card‘𝐴) ∈ ω ∧ 𝐴 ∈ ω) → ((card‘𝐴) ≈ 𝐴 ↔ (card‘𝐴) = 𝐴)) | |
| 9 | 7, 8 | mpancom 688 | . 2 ⊢ (𝐴 ∈ ω → ((card‘𝐴) ≈ 𝐴 ↔ (card‘𝐴) = 𝐴)) |
| 10 | 4, 9 | mpbid 232 | 1 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 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: card1 10008 cardennn 10023 cardsucnn 10025 nnsdomel 10030 pm54.43lem 10040 iscard3 10133 nnadju 10238 nnadjuALT 10239 ficardun 10241 ficardun2 10242 pwsdompw 10243 ackbij2 10282 sdom2en01 10342 fin23lem22 10367 fin1a2lem9 10448 ficard 10605 cfpwsdom 10624 cardfz 14011 hashgval2 14417 hashdom 14418 |
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