Theorem isfin3-3 9782

Description: Weakly Dedekind-infinite sets are exactly those with an ω-indexed descending chain of subsets. (Contributed by Stefan O'Rear, 7-Nov-2014.)

Assertion

Ref	Expression
isfin3-3	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓)))

Distinct variable groups:   𝑥,𝑓,𝐴   𝑥,𝑉

Allowed substitution hint:   𝑉(𝑓)

Proof of Theorem isfin3-3

Dummy variables 𝑎 𝑏 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isf33lem 9780	. 2 ⊢ FinIII = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) → ∩ ran 𝑎 ∈ ran 𝑎)}
2	1	isfin3ds 9743	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓)))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2108  ∀wral 3136   ⊆ wss 3934  𝒫 cpw 4537  ∩ cint 4867  ran crn 5549  suc csuc 6186  ‘cfv 6348  (class class class)co 7148  ωcom 7572   ↑_m cmap 8398  Fin^IIIcfin3 9695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-seqom 8076  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  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:  isf34lem6  9794