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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 7566 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | onenon 9362 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | |
3 | cardid2 9366 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝐴 ∈ ω → (card‘𝐴) ≈ 𝐴) |
5 | nnfi 8696 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) | |
6 | ficardom 9374 | . . . 4 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ ω → (card‘𝐴) ∈ ω) |
8 | nneneq 8684 | . . 3 ⊢ (((card‘𝐴) ∈ ω ∧ 𝐴 ∈ ω) → ((card‘𝐴) ≈ 𝐴 ↔ (card‘𝐴) = 𝐴)) | |
9 | 7, 8 | mpancom 687 | . 2 ⊢ (𝐴 ∈ ω → ((card‘𝐴) ≈ 𝐴 ↔ (card‘𝐴) = 𝐴)) |
10 | 4, 9 | mpbid 235 | 1 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 dom cdm 5519 Oncon0 6159 ‘cfv 6324 ωcom 7560 ≈ cen 8489 Fincfn 8492 cardccrd 9348 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 |
This theorem is referenced by: card1 9381 cardennn 9396 cardsucnn 9398 nnsdomel 9403 pm54.43lem 9413 iscard3 9504 nnadju 9608 nnadjuALT 9609 ficardun 9611 ficardunOLD 9612 ficardun2 9613 ficardun2OLD 9614 pwsdompw 9615 ackbij2 9654 sdom2en01 9713 fin23lem22 9738 fin1a2lem9 9819 ficard 9976 cfpwsdom 9995 cardfz 13333 hashgval2 13735 hashdom 13736 |
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