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Theorem ficard 9822
Description: A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
ficard (𝐴𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω))

Proof of Theorem ficard
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isfi 8371 . . 3 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
2 carden 9808 . . . . 5 ((𝐴𝑉𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) ↔ 𝐴𝑥))
3 cardnn 9227 . . . . . . . 8 (𝑥 ∈ ω → (card‘𝑥) = 𝑥)
4 eqtr 2814 . . . . . . . . 9 (((card‘𝐴) = (card‘𝑥) ∧ (card‘𝑥) = 𝑥) → (card‘𝐴) = 𝑥)
54expcom 414 . . . . . . . 8 ((card‘𝑥) = 𝑥 → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) = 𝑥))
63, 5syl 17 . . . . . . 7 (𝑥 ∈ ω → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) = 𝑥))
7 eleq1a 2876 . . . . . . 7 (𝑥 ∈ ω → ((card‘𝐴) = 𝑥 → (card‘𝐴) ∈ ω))
86, 7syld 47 . . . . . 6 (𝑥 ∈ ω → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) ∈ ω))
98adantl 482 . . . . 5 ((𝐴𝑉𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) ∈ ω))
102, 9sylbird 261 . . . 4 ((𝐴𝑉𝑥 ∈ ω) → (𝐴𝑥 → (card‘𝐴) ∈ ω))
1110rexlimdva 3244 . . 3 (𝐴𝑉 → (∃𝑥 ∈ ω 𝐴𝑥 → (card‘𝐴) ∈ ω))
121, 11syl5bi 243 . 2 (𝐴𝑉 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω))
13 cardnn 9227 . . . . . . . 8 ((card‘𝐴) ∈ ω → (card‘(card‘𝐴)) = (card‘𝐴))
1413eqcomd 2799 . . . . . . 7 ((card‘𝐴) ∈ ω → (card‘𝐴) = (card‘(card‘𝐴)))
1514adantl 482 . . . . . 6 ((𝐴𝑉 ∧ (card‘𝐴) ∈ ω) → (card‘𝐴) = (card‘(card‘𝐴)))
16 carden 9808 . . . . . 6 ((𝐴𝑉 ∧ (card‘𝐴) ∈ ω) → ((card‘𝐴) = (card‘(card‘𝐴)) ↔ 𝐴 ≈ (card‘𝐴)))
1715, 16mpbid 233 . . . . 5 ((𝐴𝑉 ∧ (card‘𝐴) ∈ ω) → 𝐴 ≈ (card‘𝐴))
1817ex 413 . . . 4 (𝐴𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ≈ (card‘𝐴)))
1918ancld 551 . . 3 (𝐴𝑉 → ((card‘𝐴) ∈ ω → ((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴))))
20 breq2 4960 . . . . 5 (𝑥 = (card‘𝐴) → (𝐴𝑥𝐴 ≈ (card‘𝐴)))
2120rspcev 3554 . . . 4 (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → ∃𝑥 ∈ ω 𝐴𝑥)
2221, 1sylibr 235 . . 3 (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → 𝐴 ∈ Fin)
2319, 22syl6 35 . 2 (𝐴𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ∈ Fin))
2412, 23impbid 213 1 (𝐴𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1520  wcel 2079  wrex 3104   class class class wbr 4956  cfv 6217  ωcom 7427  cen 8344  Fincfn 8347  cardccrd 9199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-ac2 9720
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-se 5395  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-isom 6226  df-riota 6968  df-om 7428  df-wrecs 7789  df-recs 7851  df-er 8130  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-card 9203  df-ac 9377
This theorem is referenced by:  cfpwsdom  9841
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