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Theorem fin33i 9783
Description: Inference from isfin3-3 9782. (This is actually a bit stronger than isfin3-3 9782 because it does not assume 𝐹 is a set and does not use the Axiom of Infinity either.) (Contributed by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
fin33i ((𝐴 ∈ FinIII𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥)) → ran 𝐹 ∈ ran 𝐹)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem fin33i
StepHypRef Expression
1 isfin32i 9779 . . 3 (𝐴 ∈ FinIII → ¬ ω ≼* 𝐴)
213ad2ant1 1127 . 2 ((𝐴 ∈ FinIII𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥)) → ¬ ω ≼* 𝐴)
3 isf32lem11 9777 . . . 4 ((𝐴 ∈ FinIII ∧ (𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥) ∧ ¬ ran 𝐹 ∈ ran 𝐹)) → ω ≼* 𝐴)
433exp2 1348 . . 3 (𝐴 ∈ FinIII → (𝐹:ω⟶𝒫 𝐴 → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥) → (¬ ran 𝐹 ∈ ran 𝐹 → ω ≼* 𝐴))))
543imp 1105 . 2 ((𝐴 ∈ FinIII𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥)) → (¬ ran 𝐹 ∈ ran 𝐹 → ω ≼* 𝐴))
62, 5mt3d 150 1 ((𝐴 ∈ FinIII𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥)) → ran 𝐹 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  w3a 1081  wcel 2106  wral 3142  wss 3939  𝒫 cpw 4541   cint 4873   class class class wbr 5062  ran crn 5554  suc csuc 6190  wf 6347  cfv 6351  ωcom 7571  * cwdom 9013  FinIIIcfin3 9695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-om 7572  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-wdom 9015  df-card 9360  df-fin4 9701  df-fin3 9702
This theorem is referenced by:  isf34lem7  9793
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