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Theorem isfiniteg 8387
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 8147 . . 3 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
2 nnsdomg 8386 . . . . 5 ((ω ∈ V ∧ 𝑥 ∈ ω) → 𝑥 ≺ ω)
3 sdomen1 8271 . . . . 5 (𝐴𝑥 → (𝐴 ≺ ω ↔ 𝑥 ≺ ω))
42, 3syl5ibrcom 237 . . . 4 ((ω ∈ V ∧ 𝑥 ∈ ω) → (𝐴𝑥𝐴 ≺ ω))
54rexlimdva 3169 . . 3 (ω ∈ V → (∃𝑥 ∈ ω 𝐴𝑥𝐴 ≺ ω))
61, 5syl5bi 232 . 2 (ω ∈ V → (𝐴 ∈ Fin → 𝐴 ≺ ω))
7 isfinite2 8385 . 2 (𝐴 ≺ ω → 𝐴 ∈ Fin)
86, 7impbid1 215 1 (ω ∈ V → (𝐴 ∈ Fin ↔ 𝐴 ≺ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2139  wrex 3051  Vcvv 3340   class class class wbr 4804  ωcom 7231  cen 8120  csdm 8122  Fincfn 8123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127
This theorem is referenced by:  unfi2  8396  unifi2  8423  isfinite  8724  axcclem  9491
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