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Theorem ficard 10367
Description: A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
ficard (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Fin ↔ (cardβ€˜π΄) ∈ Ο‰))

Proof of Theorem ficard
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 isfi 8797 . . 3 (𝐴 ∈ Fin ↔ βˆƒπ‘₯ ∈ Ο‰ 𝐴 β‰ˆ π‘₯)
2 carden 10353 . . . . 5 ((𝐴 ∈ 𝑉 ∧ π‘₯ ∈ Ο‰) β†’ ((cardβ€˜π΄) = (cardβ€˜π‘₯) ↔ 𝐴 β‰ˆ π‘₯))
3 cardnn 9765 . . . . . . . 8 (π‘₯ ∈ Ο‰ β†’ (cardβ€˜π‘₯) = π‘₯)
4 eqtr 2759 . . . . . . . . 9 (((cardβ€˜π΄) = (cardβ€˜π‘₯) ∧ (cardβ€˜π‘₯) = π‘₯) β†’ (cardβ€˜π΄) = π‘₯)
54expcom 415 . . . . . . . 8 ((cardβ€˜π‘₯) = π‘₯ β†’ ((cardβ€˜π΄) = (cardβ€˜π‘₯) β†’ (cardβ€˜π΄) = π‘₯))
63, 5syl 17 . . . . . . 7 (π‘₯ ∈ Ο‰ β†’ ((cardβ€˜π΄) = (cardβ€˜π‘₯) β†’ (cardβ€˜π΄) = π‘₯))
7 eleq1a 2832 . . . . . . 7 (π‘₯ ∈ Ο‰ β†’ ((cardβ€˜π΄) = π‘₯ β†’ (cardβ€˜π΄) ∈ Ο‰))
86, 7syld 47 . . . . . 6 (π‘₯ ∈ Ο‰ β†’ ((cardβ€˜π΄) = (cardβ€˜π‘₯) β†’ (cardβ€˜π΄) ∈ Ο‰))
98adantl 483 . . . . 5 ((𝐴 ∈ 𝑉 ∧ π‘₯ ∈ Ο‰) β†’ ((cardβ€˜π΄) = (cardβ€˜π‘₯) β†’ (cardβ€˜π΄) ∈ Ο‰))
102, 9sylbird 260 . . . 4 ((𝐴 ∈ 𝑉 ∧ π‘₯ ∈ Ο‰) β†’ (𝐴 β‰ˆ π‘₯ β†’ (cardβ€˜π΄) ∈ Ο‰))
1110rexlimdva 3149 . . 3 (𝐴 ∈ 𝑉 β†’ (βˆƒπ‘₯ ∈ Ο‰ 𝐴 β‰ˆ π‘₯ β†’ (cardβ€˜π΄) ∈ Ο‰))
121, 11biimtrid 241 . 2 (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Fin β†’ (cardβ€˜π΄) ∈ Ο‰))
13 cardnn 9765 . . . . . . . 8 ((cardβ€˜π΄) ∈ Ο‰ β†’ (cardβ€˜(cardβ€˜π΄)) = (cardβ€˜π΄))
1413eqcomd 2742 . . . . . . 7 ((cardβ€˜π΄) ∈ Ο‰ β†’ (cardβ€˜π΄) = (cardβ€˜(cardβ€˜π΄)))
1514adantl 483 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ (cardβ€˜π΄) ∈ Ο‰) β†’ (cardβ€˜π΄) = (cardβ€˜(cardβ€˜π΄)))
16 carden 10353 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ (cardβ€˜π΄) ∈ Ο‰) β†’ ((cardβ€˜π΄) = (cardβ€˜(cardβ€˜π΄)) ↔ 𝐴 β‰ˆ (cardβ€˜π΄)))
1715, 16mpbid 231 . . . . 5 ((𝐴 ∈ 𝑉 ∧ (cardβ€˜π΄) ∈ Ο‰) β†’ 𝐴 β‰ˆ (cardβ€˜π΄))
1817ex 414 . . . 4 (𝐴 ∈ 𝑉 β†’ ((cardβ€˜π΄) ∈ Ο‰ β†’ 𝐴 β‰ˆ (cardβ€˜π΄)))
1918ancld 552 . . 3 (𝐴 ∈ 𝑉 β†’ ((cardβ€˜π΄) ∈ Ο‰ β†’ ((cardβ€˜π΄) ∈ Ο‰ ∧ 𝐴 β‰ˆ (cardβ€˜π΄))))
20 breq2 5085 . . . . 5 (π‘₯ = (cardβ€˜π΄) β†’ (𝐴 β‰ˆ π‘₯ ↔ 𝐴 β‰ˆ (cardβ€˜π΄)))
2120rspcev 3566 . . . 4 (((cardβ€˜π΄) ∈ Ο‰ ∧ 𝐴 β‰ˆ (cardβ€˜π΄)) β†’ βˆƒπ‘₯ ∈ Ο‰ 𝐴 β‰ˆ π‘₯)
2221, 1sylibr 233 . . 3 (((cardβ€˜π΄) ∈ Ο‰ ∧ 𝐴 β‰ˆ (cardβ€˜π΄)) β†’ 𝐴 ∈ Fin)
2319, 22syl6 35 . 2 (𝐴 ∈ 𝑉 β†’ ((cardβ€˜π΄) ∈ Ο‰ β†’ 𝐴 ∈ Fin))
2412, 23impbid 211 1 (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Fin ↔ (cardβ€˜π΄) ∈ Ο‰))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104  βˆƒwrex 3071   class class class wbr 5081  β€˜cfv 6458  Ο‰com 7744   β‰ˆ cen 8761  Fincfn 8764  cardccrd 9737
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-ac2 10265
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-se 5556  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-isom 6467  df-riota 7264  df-ov 7310  df-om 7745  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-1o 8328  df-er 8529  df-en 8765  df-dom 8766  df-sdom 8767  df-fin 8768  df-card 9741  df-ac 9918
This theorem is referenced by:  cfpwsdom  10386
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