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Theorem ficard 10559
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 8971 . . 3 (𝐴 ∈ Fin ↔ βˆƒπ‘₯ ∈ Ο‰ 𝐴 β‰ˆ π‘₯)
2 carden 10545 . . . . 5 ((𝐴 ∈ 𝑉 ∧ π‘₯ ∈ Ο‰) β†’ ((cardβ€˜π΄) = (cardβ€˜π‘₯) ↔ 𝐴 β‰ˆ π‘₯))
3 cardnn 9957 . . . . . . . 8 (π‘₯ ∈ Ο‰ β†’ (cardβ€˜π‘₯) = π‘₯)
4 eqtr 2749 . . . . . . . . 9 (((cardβ€˜π΄) = (cardβ€˜π‘₯) ∧ (cardβ€˜π‘₯) = π‘₯) β†’ (cardβ€˜π΄) = π‘₯)
54expcom 413 . . . . . . . 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 2822 . . . . . . 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 481 . . . . 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 9957 . . . . . . . 8 ((cardβ€˜π΄) ∈ Ο‰ β†’ (cardβ€˜(cardβ€˜π΄)) = (cardβ€˜π΄))
1413eqcomd 2732 . . . . . . 7 ((cardβ€˜π΄) ∈ Ο‰ β†’ (cardβ€˜π΄) = (cardβ€˜(cardβ€˜π΄)))
1514adantl 481 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ (cardβ€˜π΄) ∈ Ο‰) β†’ (cardβ€˜π΄) = (cardβ€˜(cardβ€˜π΄)))
16 carden 10545 . . . . . 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 412 . . . 4 (𝐴 ∈ 𝑉 β†’ ((cardβ€˜π΄) ∈ Ο‰ β†’ 𝐴 β‰ˆ (cardβ€˜π΄)))
1918ancld 550 . . 3 (𝐴 ∈ 𝑉 β†’ ((cardβ€˜π΄) ∈ Ο‰ β†’ ((cardβ€˜π΄) ∈ Ο‰ ∧ 𝐴 β‰ˆ (cardβ€˜π΄))))
20 breq2 5145 . . . . 5 (π‘₯ = (cardβ€˜π΄) β†’ (𝐴 β‰ˆ π‘₯ ↔ 𝐴 β‰ˆ (cardβ€˜π΄)))
2120rspcev 3606 . . . 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 395   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064   class class class wbr 5141  β€˜cfv 6536  Ο‰com 7851   β‰ˆ cen 8935  Fincfn 8938  cardccrd 9929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-ac2 10457
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7360  df-ov 7407  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-1o 8464  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-ac 10110
This theorem is referenced by:  cfpwsdom  10578
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