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Theorem isfiniteg 8766
Description: A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 3-Nov-2002.) (Revised by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
isfiniteg (ω ∈ V → (𝐴 ∈ Fin ↔ 𝐴 ≺ ω))

Proof of Theorem isfiniteg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isfi 8520 . . 3 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
2 nnsdomg 8765 . . . . 5 ((ω ∈ V ∧ 𝑥 ∈ ω) → 𝑥 ≺ ω)
3 sdomen1 8649 . . . . 5 (𝐴𝑥 → (𝐴 ≺ ω ↔ 𝑥 ≺ ω))
42, 3syl5ibrcom 250 . . . 4 ((ω ∈ V ∧ 𝑥 ∈ ω) → (𝐴𝑥𝐴 ≺ ω))
54rexlimdva 3270 . . 3 (ω ∈ V → (∃𝑥 ∈ ω 𝐴𝑥𝐴 ≺ ω))
61, 5syl5bi 245 . 2 (ω ∈ V → (𝐴 ∈ Fin → 𝐴 ≺ ω))
7 isfinite2 8764 . 2 (𝐴 ≺ ω → 𝐴 ∈ Fin)
86, 7impbid1 228 1 (ω ∈ V → (𝐴 ∈ Fin ↔ 𝐴 ≺ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2114  wrex 3131  Vcvv 3469   class class class wbr 5042  ωcom 7565  cen 8493  csdm 8495  Fincfn 8496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500
This theorem is referenced by:  unfi2  8775  unifi2  8802  isfinite  9103  axcclem  9868
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