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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 7589 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | onenon 9381 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | |
3 | cardid2 9385 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝐴 ∈ ω → (card‘𝐴) ≈ 𝐴) |
5 | nnfi 8714 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) | |
6 | ficardom 9393 | . . . 4 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ ω → (card‘𝐴) ∈ ω) |
8 | nneneq 8703 | . . 3 ⊢ (((card‘𝐴) ∈ ω ∧ 𝐴 ∈ ω) → ((card‘𝐴) ≈ 𝐴 ↔ (card‘𝐴) = 𝐴)) | |
9 | 7, 8 | mpancom 686 | . 2 ⊢ (𝐴 ∈ ω → ((card‘𝐴) ≈ 𝐴 ↔ (card‘𝐴) = 𝐴)) |
10 | 4, 9 | mpbid 234 | 1 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 dom cdm 5558 Oncon0 6194 ‘cfv 6358 ωcom 7583 ≈ cen 8509 Fincfn 8512 cardccrd 9367 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7584 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-card 9371 |
This theorem is referenced by: card1 9400 cardennn 9415 cardsucnn 9417 nnsdomel 9422 pm54.43lem 9431 iscard3 9522 nnadju 9626 ficardun 9627 ficardun2 9628 pwsdompw 9629 ackbij2 9668 sdom2en01 9727 fin23lem22 9752 fin1a2lem9 9833 ficard 9990 cfpwsdom 10009 cardfz 13341 hashgval2 13742 hashdom 13743 |
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