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Theorem onfin 8036
 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 7865 . 2 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
2 onomeneq 8035 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴𝑥𝐴 = 𝑥))
3 eleq1a 2683 . . . . . 6 (𝑥 ∈ ω → (𝐴 = 𝑥𝐴 ∈ ω))
43adantl 481 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 = 𝑥𝐴 ∈ ω))
52, 4sylbid 229 . . . 4 ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴𝑥𝐴 ∈ ω))
65rexlimdva 3013 . . 3 (𝐴 ∈ On → (∃𝑥 ∈ ω 𝐴𝑥𝐴 ∈ ω))
7 enrefg 7873 . . . 4 (𝐴 ∈ ω → 𝐴𝐴)
8 breq2 4587 . . . . 5 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
98rspcev 3282 . . . 4 ((𝐴 ∈ ω ∧ 𝐴𝐴) → ∃𝑥 ∈ ω 𝐴𝑥)
107, 9mpdan 699 . . 3 (𝐴 ∈ ω → ∃𝑥 ∈ ω 𝐴𝑥)
116, 10impbid1 214 . 2 (𝐴 ∈ On → (∃𝑥 ∈ ω 𝐴𝑥𝐴 ∈ ω))
121, 11syl5bb 271 1 (𝐴 ∈ On → (𝐴 ∈ Fin ↔ 𝐴 ∈ ω))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   class class class wbr 4583  Oncon0 5640  ωcom 6957   ≈ cen 7838  Fincfn 7841 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845 This theorem is referenced by:  onfin2  8037  fin17  9099  isfin7-2  9101
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