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Theorem isfiniteg 9334
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 9014 . . 3 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
2 nnsdomg 9332 . . . . 5 ((ω ∈ V ∧ 𝑥 ∈ ω) → 𝑥 ≺ ω)
3 sdomen1 9159 . . . . 5 (𝐴𝑥 → (𝐴 ≺ ω ↔ 𝑥 ≺ ω))
42, 3syl5ibrcom 247 . . . 4 ((ω ∈ V ∧ 𝑥 ∈ ω) → (𝐴𝑥𝐴 ≺ ω))
54rexlimdva 3152 . . 3 (ω ∈ V → (∃𝑥 ∈ ω 𝐴𝑥𝐴 ≺ ω))
61, 5biimtrid 242 . 2 (ω ∈ V → (𝐴 ∈ Fin → 𝐴 ≺ ω))
7 isfinite2 9331 . 2 (𝐴 ≺ ω → 𝐴 ∈ Fin)
86, 7impbid1 225 1 (ω ∈ V → (𝐴 ∈ Fin ↔ 𝐴 ≺ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2105  wrex 3067  Vcvv 3477   class class class wbr 5147  ωcom 7886  cen 8980  csdm 8982  Fincfn 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987
This theorem is referenced by:  unfi2  9345  unifi2  9382  isfinite  9689  axcclem  10494
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