MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin33i Structured version   Visualization version   GIF version

Theorem fin33i 9480
Description: Inference from isfin3-3 9479. (This is actually a bit stronger than isfin3-3 9479 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 9476 . . 3 (𝐴 ∈ FinIII → ¬ ω ≼* 𝐴)
213ad2ant1 1164 . 2 ((𝐴 ∈ FinIII𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥)) → ¬ ω ≼* 𝐴)
3 isf32lem11 9474 . . . 4 ((𝐴 ∈ FinIII ∧ (𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥) ∧ ¬ ran 𝐹 ∈ ran 𝐹)) → ω ≼* 𝐴)
433exp2 1464 . . 3 (𝐴 ∈ FinIII → (𝐹:ω⟶𝒫 𝐴 → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥) → (¬ ran 𝐹 ∈ ran 𝐹 → ω ≼* 𝐴))))
543imp 1138 . 2 ((𝐴 ∈ FinIII𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥)) → (¬ ran 𝐹 ∈ ran 𝐹 → ω ≼* 𝐴))
62, 5mt3d 143 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 1108  wcel 2157  wral 3090  wss 3770  𝒫 cpw 4350   cint 4668   class class class wbr 4844  ran crn 5314  suc csuc 5944  wf 6098  cfv 6102  ωcom 7300  * cwdom 8705  FinIIIcfin3 9392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-int 4669  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5221  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-se 5273  df-we 5274  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-pred 5899  df-ord 5945  df-on 5946  df-lim 5947  df-suc 5948  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-isom 6111  df-riota 6840  df-om 7301  df-wrecs 7646  df-recs 7708  df-rdg 7746  df-1o 7800  df-er 7983  df-en 8197  df-dom 8198  df-sdom 8199  df-fin 8200  df-wdom 8707  df-card 9052  df-fin4 9398  df-fin3 9399
This theorem is referenced by:  isf34lem7  9490
  Copyright terms: Public domain W3C validator