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Theorem onfin 8697
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 8521 . 2 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
2 onomeneq 8696 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴𝑥𝐴 = 𝑥))
3 eleq1a 2905 . . . . . 6 (𝑥 ∈ ω → (𝐴 = 𝑥𝐴 ∈ ω))
43adantl 482 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 = 𝑥𝐴 ∈ ω))
52, 4sylbid 241 . . . 4 ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴𝑥𝐴 ∈ ω))
65rexlimdva 3281 . . 3 (𝐴 ∈ On → (∃𝑥 ∈ ω 𝐴𝑥𝐴 ∈ ω))
7 enrefg 8529 . . . 4 (𝐴 ∈ ω → 𝐴𝐴)
8 breq2 5061 . . . . 5 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
98rspcev 3620 . . . 4 ((𝐴 ∈ ω ∧ 𝐴𝐴) → ∃𝑥 ∈ ω 𝐴𝑥)
107, 9mpdan 683 . . 3 (𝐴 ∈ ω → ∃𝑥 ∈ ω 𝐴𝑥)
116, 10impbid1 226 . 2 (𝐴 ∈ On → (∃𝑥 ∈ ω 𝐴𝑥𝐴 ∈ ω))
121, 11syl5bb 284 1 (𝐴 ∈ On → (𝐴 ∈ Fin ↔ 𝐴 ∈ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3136   class class class wbr 5057  Oncon0 6184  ωcom 7569  cen 8494  Fincfn 8497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501
This theorem is referenced by:  onfin2  8698  fin17  9804  isfin7-2  9806
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