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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 7349 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | onenon 9108 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | |
3 | cardid2 9112 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝐴 ∈ ω → (card‘𝐴) ≈ 𝐴) |
5 | nnfi 8441 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) | |
6 | ficardom 9120 | . . . 4 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ ω → (card‘𝐴) ∈ ω) |
8 | nneneq 8431 | . . 3 ⊢ (((card‘𝐴) ∈ ω ∧ 𝐴 ∈ ω) → ((card‘𝐴) ≈ 𝐴 ↔ (card‘𝐴) = 𝐴)) | |
9 | 7, 8 | mpancom 678 | . 2 ⊢ (𝐴 ∈ ω → ((card‘𝐴) ≈ 𝐴 ↔ (card‘𝐴) = 𝐴)) |
10 | 4, 9 | mpbid 224 | 1 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 class class class wbr 4886 dom cdm 5355 Oncon0 5976 ‘cfv 6135 ωcom 7343 ≈ cen 8238 Fincfn 8241 cardccrd 9094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-om 7344 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 |
This theorem is referenced by: card1 9127 cardennn 9142 cardsucnn 9144 nnsdomel 9149 pm54.43lem 9158 iscard3 9249 nnacda 9358 ficardun 9359 ficardun2 9360 pwsdompw 9361 ackbij2 9400 sdom2en01 9459 fin23lem22 9484 fin1a2lem9 9565 ficard 9722 cfpwsdom 9741 cardfz 13088 hashgval2 13482 hashdom 13483 |
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