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Theorem ficard 9976
 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 8516 . . 3 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
2 carden 9962 . . . . 5 ((𝐴𝑉𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) ↔ 𝐴𝑥))
3 cardnn 9376 . . . . . . . 8 (𝑥 ∈ ω → (card‘𝑥) = 𝑥)
4 eqtr 2818 . . . . . . . . 9 (((card‘𝐴) = (card‘𝑥) ∧ (card‘𝑥) = 𝑥) → (card‘𝐴) = 𝑥)
54expcom 417 . . . . . . . 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 2885 . . . . . . 7 (𝑥 ∈ ω → ((card‘𝐴) = 𝑥 → (card‘𝐴) ∈ ω))
86, 7syld 47 . . . . . 6 (𝑥 ∈ ω → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) ∈ ω))
98adantl 485 . . . . 5 ((𝐴𝑉𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) ∈ ω))
102, 9sylbird 263 . . . 4 ((𝐴𝑉𝑥 ∈ ω) → (𝐴𝑥 → (card‘𝐴) ∈ ω))
1110rexlimdva 3243 . . 3 (𝐴𝑉 → (∃𝑥 ∈ ω 𝐴𝑥 → (card‘𝐴) ∈ ω))
121, 11syl5bi 245 . 2 (𝐴𝑉 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω))
13 cardnn 9376 . . . . . . . 8 ((card‘𝐴) ∈ ω → (card‘(card‘𝐴)) = (card‘𝐴))
1413eqcomd 2804 . . . . . . 7 ((card‘𝐴) ∈ ω → (card‘𝐴) = (card‘(card‘𝐴)))
1514adantl 485 . . . . . 6 ((𝐴𝑉 ∧ (card‘𝐴) ∈ ω) → (card‘𝐴) = (card‘(card‘𝐴)))
16 carden 9962 . . . . . 6 ((𝐴𝑉 ∧ (card‘𝐴) ∈ ω) → ((card‘𝐴) = (card‘(card‘𝐴)) ↔ 𝐴 ≈ (card‘𝐴)))
1715, 16mpbid 235 . . . . 5 ((𝐴𝑉 ∧ (card‘𝐴) ∈ ω) → 𝐴 ≈ (card‘𝐴))
1817ex 416 . . . 4 (𝐴𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ≈ (card‘𝐴)))
1918ancld 554 . . 3 (𝐴𝑉 → ((card‘𝐴) ∈ ω → ((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴))))
20 breq2 5034 . . . . 5 (𝑥 = (card‘𝐴) → (𝐴𝑥𝐴 ≈ (card‘𝐴)))
2120rspcev 3571 . . . 4 (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → ∃𝑥 ∈ ω 𝐴𝑥)
2221, 1sylibr 237 . . 3 (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → 𝐴 ∈ Fin)
2319, 22syl6 35 . 2 (𝐴𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ∈ Fin))
2412, 23impbid 215 1 (𝐴𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   class class class wbr 5030  ‘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-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-ac2 9874 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-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  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-iun 4883  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-se 5479  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-pred 6116  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-isom 6333  df-riota 7093  df-om 7561  df-wrecs 7930  df-recs 7991  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-card 9352  df-ac 9527 This theorem is referenced by:  cfpwsdom  9995
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