isfin3-2 - Metamath Proof Explorer

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Theorem isfin3-2 9827

Description: Weakly Dedekind-infinite sets are exactly those which can be mapped onto ω. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)

**Assertion**
Ref	Expression
isfin3-2	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin^III ↔ ¬ ω ≼^* 𝐴))

Proof of Theorem isfin3-2 Dummy variables 𝑎 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfin32i 9825	. 2 ⊢ (𝐴 ∈ Fin^III → ¬ ω ≼^* 𝐴)
2		isf33lem 9826	. . 3 ⊢ Fin^III = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑_m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
3	2	isf32lem12 9824	. 2 ⊢ (𝐴 ∈ 𝑉 → (¬ ω ≼^* 𝐴 → 𝐴 ∈ Fin^III))
4	1, 3	impbid2 229	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin^III ↔ ¬ ω ≼^* 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∈ wcel 2111 class class class wbr 5032 ωcom 7579 ≼^* cwdom 9061 Fin^IIIcfin3 9741

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5156 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-pss 3877 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-tp 4527 df-op 4529 df-uni 4799 df-int 4839 df-iun 4885 df-br 5033 df-opab 5095 df-mpt 5113 df-tr 5139 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5483 df-se 5484 df-we 5485 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-pred 6126 df-ord 6172 df-on 6173 df-lim 6174 df-suc 6175 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-isom 6344 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7580 df-1st 7693 df-2nd 7694 df-wrecs 7957 df-recs 8018 df-rdg 8056 df-seqom 8094 df-1o 8112 df-er 8299 df-map 8418 df-en 8528 df-dom 8529 df-sdom 8530 df-fin 8531 df-wdom 9062 df-card 9401 df-fin4 9747 df-fin3 9748

This theorem is referenced by: fin1a2lem7 9866

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