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Mirrors > Home > MPE Home > Th. List > isfin3-3 | Structured version Visualization version GIF version |
Description: Weakly Dedekind-infinite sets are exactly those with an ω-indexed descending chain of subsets. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Ref | Expression |
---|---|
isfin3-3 | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑𝑚 ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isf33lem 9071 | . 2 ⊢ FinIII = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑𝑚 ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) → ∩ ran 𝑎 ∈ ran 𝑎)} | |
2 | 1 | isfin3ds 9034 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑𝑚 ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 𝒫 cpw 4108 ∩ cint 4410 ran crn 5039 suc csuc 5642 ‘cfv 5804 (class class class)co 6549 ωcom 6957 ↑𝑚 cmap 7744 FinIIIcfin3 8986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seqom 7430 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-wdom 8347 df-card 8648 df-fin4 8992 df-fin3 8993 |
This theorem is referenced by: isf34lem6 9085 |
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