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Theorem ficard 9331
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 7923 . . 3 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
2 carden 9317 . . . . 5 ((𝐴𝑉𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) ↔ 𝐴𝑥))
3 cardnn 8733 . . . . . . . 8 (𝑥 ∈ ω → (card‘𝑥) = 𝑥)
4 eqtr 2640 . . . . . . . . 9 (((card‘𝐴) = (card‘𝑥) ∧ (card‘𝑥) = 𝑥) → (card‘𝐴) = 𝑥)
54expcom 451 . . . . . . . 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 2693 . . . . . . 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 250 . . . 4 ((𝐴𝑉𝑥 ∈ ω) → (𝐴𝑥 → (card‘𝐴) ∈ ω))
1110rexlimdva 3024 . . 3 (𝐴𝑉 → (∃𝑥 ∈ ω 𝐴𝑥 → (card‘𝐴) ∈ ω))
121, 11syl5bi 232 . 2 (𝐴𝑉 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω))
13 cardnn 8733 . . . . . . . 8 ((card‘𝐴) ∈ ω → (card‘(card‘𝐴)) = (card‘𝐴))
1413eqcomd 2627 . . . . . . 7 ((card‘𝐴) ∈ ω → (card‘𝐴) = (card‘(card‘𝐴)))
1514adantl 482 . . . . . 6 ((𝐴𝑉 ∧ (card‘𝐴) ∈ ω) → (card‘𝐴) = (card‘(card‘𝐴)))
16 carden 9317 . . . . . 6 ((𝐴𝑉 ∧ (card‘𝐴) ∈ ω) → ((card‘𝐴) = (card‘(card‘𝐴)) ↔ 𝐴 ≈ (card‘𝐴)))
1715, 16mpbid 222 . . . . 5 ((𝐴𝑉 ∧ (card‘𝐴) ∈ ω) → 𝐴 ≈ (card‘𝐴))
1817ex 450 . . . 4 (𝐴𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ≈ (card‘𝐴)))
1918ancld 575 . . 3 (𝐴𝑉 → ((card‘𝐴) ∈ ω → ((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴))))
20 breq2 4617 . . . . 5 (𝑥 = (card‘𝐴) → (𝐴𝑥𝐴 ≈ (card‘𝐴)))
2120rspcev 3295 . . . 4 (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → ∃𝑥 ∈ ω 𝐴𝑥)
2221, 1sylibr 224 . . 3 (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → 𝐴 ∈ Fin)
2319, 22syl6 35 . 2 (𝐴𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ∈ Fin))
2412, 23impbid 202 1 (𝐴𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908   class class class wbr 4613  cfv 5847  ωcom 7012  cen 7896  Fincfn 7899  cardccrd 8705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-ac2 9229
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-om 7013  df-wrecs 7352  df-recs 7413  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-ac 8883
This theorem is referenced by:  cfpwsdom  9350
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