MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ficard Structured version   Visualization version   GIF version

Theorem ficard 10590
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 8997 . . 3 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
2 carden 10576 . . . . 5 ((𝐴𝑉𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) ↔ 𝐴𝑥))
3 cardnn 9988 . . . . . . . 8 (𝑥 ∈ ω → (card‘𝑥) = 𝑥)
4 eqtr 2748 . . . . . . . . 9 (((card‘𝐴) = (card‘𝑥) ∧ (card‘𝑥) = 𝑥) → (card‘𝐴) = 𝑥)
54expcom 412 . . . . . . . 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 2820 . . . . . . 7 (𝑥 ∈ ω → ((card‘𝐴) = 𝑥 → (card‘𝐴) ∈ ω))
86, 7syld 47 . . . . . 6 (𝑥 ∈ ω → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) ∈ ω))
98adantl 480 . . . . 5 ((𝐴𝑉𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) ∈ ω))
102, 9sylbird 259 . . . 4 ((𝐴𝑉𝑥 ∈ ω) → (𝐴𝑥 → (card‘𝐴) ∈ ω))
1110rexlimdva 3144 . . 3 (𝐴𝑉 → (∃𝑥 ∈ ω 𝐴𝑥 → (card‘𝐴) ∈ ω))
121, 11biimtrid 241 . 2 (𝐴𝑉 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω))
13 cardnn 9988 . . . . . . . 8 ((card‘𝐴) ∈ ω → (card‘(card‘𝐴)) = (card‘𝐴))
1413eqcomd 2731 . . . . . . 7 ((card‘𝐴) ∈ ω → (card‘𝐴) = (card‘(card‘𝐴)))
1514adantl 480 . . . . . 6 ((𝐴𝑉 ∧ (card‘𝐴) ∈ ω) → (card‘𝐴) = (card‘(card‘𝐴)))
16 carden 10576 . . . . . 6 ((𝐴𝑉 ∧ (card‘𝐴) ∈ ω) → ((card‘𝐴) = (card‘(card‘𝐴)) ↔ 𝐴 ≈ (card‘𝐴)))
1715, 16mpbid 231 . . . . 5 ((𝐴𝑉 ∧ (card‘𝐴) ∈ ω) → 𝐴 ≈ (card‘𝐴))
1817ex 411 . . . 4 (𝐴𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ≈ (card‘𝐴)))
1918ancld 549 . . 3 (𝐴𝑉 → ((card‘𝐴) ∈ ω → ((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴))))
20 breq2 5153 . . . . 5 (𝑥 = (card‘𝐴) → (𝐴𝑥𝐴 ≈ (card‘𝐴)))
2120rspcev 3606 . . . 4 (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → ∃𝑥 ∈ ω 𝐴𝑥)
2221, 1sylibr 233 . . 3 (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → 𝐴 ∈ Fin)
2319, 22syl6 35 . 2 (𝐴𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ∈ Fin))
2412, 23impbid 211 1 (𝐴𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wrex 3059   class class class wbr 5149  cfv 6549  ωcom 7871  cen 8961  Fincfn 8964  cardccrd 9960
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 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-ac2 10488
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9964  df-ac 10141
This theorem is referenced by:  cfpwsdom  10609
  Copyright terms: Public domain W3C validator