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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 7892 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | onenon 9986 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | |
3 | cardid2 9990 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝐴 ∈ ω → (card‘𝐴) ≈ 𝐴) |
5 | nnfi 9205 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) | |
6 | ficardom 9998 | . . . 4 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ ω → (card‘𝐴) ∈ ω) |
8 | nneneq 9243 | . . 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 1536 ∈ wcel 2105 class class class wbr 5147 dom cdm 5688 Oncon0 6385 ‘cfv 6562 ωcom 7886 ≈ cen 8980 Fincfn 8983 cardccrd 9972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-om 7887 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-card 9976 |
This theorem is referenced by: card1 10005 cardennn 10020 cardsucnn 10022 nnsdomel 10027 pm54.43lem 10037 iscard3 10130 nnadju 10235 nnadjuALT 10236 ficardun 10238 ficardun2 10239 pwsdompw 10240 ackbij2 10279 sdom2en01 10339 fin23lem22 10364 fin1a2lem9 10445 ficard 10602 cfpwsdom 10621 cardfz 14007 hashgval2 14413 hashdom 14414 |
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