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Theorem isfiniteg 9255
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 8923 . . 3 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
2 nnsdomg 9253 . . . . 5 ((ω ∈ V ∧ 𝑥 ∈ ω) → 𝑥 ≺ ω)
3 sdomen1 9072 . . . . 5 (𝐴𝑥 → (𝐴 ≺ ω ↔ 𝑥 ≺ ω))
42, 3syl5ibrcom 247 . . . 4 ((ω ∈ V ∧ 𝑥 ∈ ω) → (𝐴𝑥𝐴 ≺ ω))
54rexlimdva 3153 . . 3 (ω ∈ V → (∃𝑥 ∈ ω 𝐴𝑥𝐴 ≺ ω))
61, 5biimtrid 241 . 2 (ω ∈ V → (𝐴 ∈ Fin → 𝐴 ≺ ω))
7 isfinite2 9252 . 2 (𝐴 ≺ ω → 𝐴 ∈ Fin)
86, 7impbid1 224 1 (ω ∈ V → (𝐴 ∈ Fin ↔ 𝐴 ≺ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  wrex 3074  Vcvv 3448   class class class wbr 5110  ωcom 7807  cen 8887  csdm 8889  Fincfn 8890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894
This theorem is referenced by:  unfi2  9266  unifi2  9293  isfinite  9595  axcclem  10400
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