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Theorem onfin 8789
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 8579 . 2 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
2 onomeneq 8788 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴𝑥𝐴 = 𝑥))
3 eleq1a 2828 . . . . . 6 (𝑥 ∈ ω → (𝐴 = 𝑥𝐴 ∈ ω))
43adantl 485 . . . . 5 ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 = 𝑥𝐴 ∈ ω))
52, 4sylbid 243 . . . 4 ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴𝑥𝐴 ∈ ω))
65rexlimdva 3194 . . 3 (𝐴 ∈ On → (∃𝑥 ∈ ω 𝐴𝑥𝐴 ∈ ω))
7 enrefg 8587 . . . 4 (𝐴 ∈ ω → 𝐴𝐴)
8 breq2 5034 . . . . 5 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
98rspcev 3526 . . . 4 ((𝐴 ∈ ω ∧ 𝐴𝐴) → ∃𝑥 ∈ ω 𝐴𝑥)
107, 9mpdan 687 . . 3 (𝐴 ∈ ω → ∃𝑥 ∈ ω 𝐴𝑥)
116, 10impbid1 228 . 2 (𝐴 ∈ On → (∃𝑥 ∈ ω 𝐴𝑥𝐴 ∈ ω))
121, 11syl5bb 286 1 (𝐴 ∈ On → (𝐴 ∈ Fin ↔ 𝐴 ∈ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wrex 3054   class class class wbr 5030  Oncon0 6172  ωcom 7599  cen 8552  Fincfn 8555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7600  df-er 8320  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559
This theorem is referenced by:  onfin2  8790  fin17  9894  isfin7-2  9896
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