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Theorem onfin 8192
Description: An ordinal number is finite iff it is a natural number. Proposition 10.32 of [TakeutiZaring] p. 92. (Contributed by NM, 26-Jul-2004.)
Assertion
Ref Expression
onfin (𝐴 ∈ On → (𝐴 ∈ Fin ↔ 𝐴 ∈ ω))

Proof of Theorem onfin
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isfi 8021 . 2 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
2 onomeneq 8191 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴𝑥𝐴 = 𝑥))
3 eleq1a 2725 . . . . . 6 (𝑥 ∈ ω → (𝐴 = 𝑥𝐴 ∈ ω))
43adantl 481 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 = 𝑥𝐴 ∈ ω))
52, 4sylbid 230 . . . 4 ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴𝑥𝐴 ∈ ω))
65rexlimdva 3060 . . 3 (𝐴 ∈ On → (∃𝑥 ∈ ω 𝐴𝑥𝐴 ∈ ω))
7 enrefg 8029 . . . 4 (𝐴 ∈ ω → 𝐴𝐴)
8 breq2 4689 . . . . 5 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
98rspcev 3340 . . . 4 ((𝐴 ∈ ω ∧ 𝐴𝐴) → ∃𝑥 ∈ ω 𝐴𝑥)
107, 9mpdan 703 . . 3 (𝐴 ∈ ω → ∃𝑥 ∈ ω 𝐴𝑥)
116, 10impbid1 215 . 2 (𝐴 ∈ On → (∃𝑥 ∈ ω 𝐴𝑥𝐴 ∈ ω))
121, 11syl5bb 272 1 (𝐴 ∈ On → (𝐴 ∈ Fin ↔ 𝐴 ∈ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942   class class class wbr 4685  Oncon0 5761  ωcom 7107  cen 7994  Fincfn 7997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001
This theorem is referenced by:  onfin2  8193  fin17  9254  isfin7-2  9256
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