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Theorem ficard 10594
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 9001 . . 3 (𝐴 ∈ Fin ↔ βˆƒπ‘₯ ∈ Ο‰ 𝐴 β‰ˆ π‘₯)
2 carden 10580 . . . . 5 ((𝐴 ∈ 𝑉 ∧ π‘₯ ∈ Ο‰) β†’ ((cardβ€˜π΄) = (cardβ€˜π‘₯) ↔ 𝐴 β‰ˆ π‘₯))
3 cardnn 9992 . . . . . . . 8 (π‘₯ ∈ Ο‰ β†’ (cardβ€˜π‘₯) = π‘₯)
4 eqtr 2750 . . . . . . . . 9 (((cardβ€˜π΄) = (cardβ€˜π‘₯) ∧ (cardβ€˜π‘₯) = π‘₯) β†’ (cardβ€˜π΄) = π‘₯)
54expcom 412 . . . . . . . 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 2823 . . . . . . 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 480 . . . . 5 ((𝐴 ∈ 𝑉 ∧ π‘₯ ∈ Ο‰) β†’ ((cardβ€˜π΄) = (cardβ€˜π‘₯) β†’ (cardβ€˜π΄) ∈ Ο‰))
102, 9sylbird 259 . . . 4 ((𝐴 ∈ 𝑉 ∧ π‘₯ ∈ Ο‰) β†’ (𝐴 β‰ˆ π‘₯ β†’ (cardβ€˜π΄) ∈ Ο‰))
1110rexlimdva 3151 . . 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 9992 . . . . . . . 8 ((cardβ€˜π΄) ∈ Ο‰ β†’ (cardβ€˜(cardβ€˜π΄)) = (cardβ€˜π΄))
1413eqcomd 2733 . . . . . . 7 ((cardβ€˜π΄) ∈ Ο‰ β†’ (cardβ€˜π΄) = (cardβ€˜(cardβ€˜π΄)))
1514adantl 480 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ (cardβ€˜π΄) ∈ Ο‰) β†’ (cardβ€˜π΄) = (cardβ€˜(cardβ€˜π΄)))
16 carden 10580 . . . . . 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 411 . . . 4 (𝐴 ∈ 𝑉 β†’ ((cardβ€˜π΄) ∈ Ο‰ β†’ 𝐴 β‰ˆ (cardβ€˜π΄)))
1918ancld 549 . . 3 (𝐴 ∈ 𝑉 β†’ ((cardβ€˜π΄) ∈ Ο‰ β†’ ((cardβ€˜π΄) ∈ Ο‰ ∧ 𝐴 β‰ˆ (cardβ€˜π΄))))
20 breq2 5154 . . . . 5 (π‘₯ = (cardβ€˜π΄) β†’ (𝐴 β‰ˆ π‘₯ ↔ 𝐴 β‰ˆ (cardβ€˜π΄)))
2120rspcev 3609 . . . 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 394   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3066   class class class wbr 5150  β€˜cfv 6551  Ο‰com 7874   β‰ˆ cen 8965  Fincfn 8968  cardccrd 9964
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 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-ac2 10492
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-se 5636  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-isom 6560  df-riota 7380  df-ov 7427  df-om 7875  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-1o 8491  df-er 8729  df-en 8969  df-dom 8970  df-sdom 8971  df-fin 8972  df-card 9968  df-ac 10145
This theorem is referenced by:  cfpwsdom  10613
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